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Symbolic Integration

 

Indefinite integration

Definite integration

Indefinite integration

The capabilities of finding indefinite integrals in Maple have been improved. The following integrals could not be computed in previous versions of Maple. In particular, this applies to many integrands involving inverse hyperbolic functions, such as the following:

∫arcsinhxx ⅆx

−12⁢arcsinh⁡x2+arcsinh⁡x⁢ln⁡1−x−x2+1+polylog⁡2,x+x2+1+arcsinh⁡x⁢ln⁡1+x+x2+1+polylog⁡2,−x−x2+1

(1.1)

ⅆⅆx

−arcsinh⁡xx2+1+ln⁡1−x−x2+1x2+1+arcsinh⁡x⁢−1−xx2+11−x−x2+1−1+xx2+1⁢ln⁡1−x−x2+1x+x2+1+ln⁡1+x+x2+1x2+1+arcsinh⁡x⁢1+xx2+11+x+x2+1−−1−xx2+1⁢ln⁡1+x+x2+1−x−x2+1

(1.2)

radnormal

arcsinh⁡xx

(1.3)

∫arccothx3ⅆx

arccoth⁡x3⁢x−1+2⁢arccoth⁡x3−3⁢arccoth⁡x2⁢ln⁡1−1x−1x+1−6⁢arccoth⁡x⁢polylog⁡2,1x−1x+1+6⁢polylog⁡3,1x−1x+1−3⁢arccoth⁡x2⁢ln⁡1+1x−1x+1−6⁢arccoth⁡x⁢polylog⁡2,−1x−1x+1+6⁢polylog⁡3,−1x−1x+1

(1.4)

radnormalⅆⅆx

arccoth⁡x3

(1.5)

∫arctanhtanhb x+a2x ⅆx

ln⁡x⁢arctanh⁡tanh⁡b⁢x+a2+b2⁢x2⁢ln⁡x−32⁢b2⁢x2−2⁢b⁢ln⁡x⁢arctanh⁡tanh⁡b⁢x+a⁢x+2⁢b⁢arctanh⁡tanh⁡b⁢x+a⁢x

(1.6)

ⅆⅆx

arctanh⁡tanh⁡b⁢x+a2x

(1.7)

Some other types of integrands are covered by the improvements as well.

∫ln⁡x+an2x+b2ⅆx

−ln⁡x+an2x+b+2⁢n⁢ln⁡x+an⁢ln⁡x+ba−b−2⁢n⁢ln⁡x+an⁢ln⁡x+aa−b−2⁢n2⁢ln⁡x+b⁢ln⁡x+aa−ba−b−2⁢n2⁢dilog⁡x+aa−ba−b+n2⁢ln⁡x+a2a−b

(1.8)

normalⅆⅆx

ln⁡x+an2x+b2

(1.9)

More compact results

Some integrals that used to be expressed in terms of lengthy csgn expressions are now are given in more compact form.

∫x arctantanx3ⅆx

12⁢x2⁢arctan⁡tan⁡x3−12⁢x3⁢arctan⁡tan⁡x2+14⁢x4⁢arctan⁡tan⁡x−120⁢x5

(1.1.1)

ⅆⅆx

x⁢arctan⁡tan⁡x3

(1.1.2)

Definite integration

Definite integrals can now also be computed for some non-smooth integrands, for which previous versions of Maple could only compute an indefinite integral.

∫−∞∞x−11/3x2+1ⅆx

−3⁢∑_R=RootOf⁡23328⁢_Z6+216⁢_Z3+1_R⁢ln⁡6+_R⁢ln⁡−216⁢_R4−_R−3⁢∑_R=RootOf⁡23328⁢_Z6+216⁢_Z3+1_R⁢ln⁡6+_R⁢ln⁡216⁢_R4+_R

(2.1)

radnormalevalcallvalues

16⁢22/3⁢π⁢3+3

(2.2)