3.3 Maple 2020 and Maple 2019.2.1

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3.3.1 Test number 55

Test folder name

test_cases/2_Exponentials/2.3_Exponential_functions

3.3.1.1 Problem number 591

\[ \int \frac{F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx \] Optimal antiderivative

\[ \frac{\sqrt{\pi } F^{a f} \text{Erfi}\left (\sqrt{b} \sqrt{f} \sqrt{\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt{b} e \sqrt{f} g n \sqrt{\log (F)}} \] command

int(F^(f*(a+b*ln(c*(e*x+d)^n)^2))/(e*g*x+d*g),x)

Maple 2020 output

\[ \int{\frac{{F}^{f \left ( a+b \left ( \ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \right ) }}{egx+dg}}\, dx \] Maple 2019.2.1 output

\[{\frac{\sqrt{\pi }{F}^{af}}{2\,neg}{\it Erf} \left ( \sqrt{-\ln \left ( F \right ) bf}\ln \left ( \left ( ex+d \right ) ^{n} \right ) -{\frac{bf \left ( 2\,\ln \left ( c \right ) -i\pi \,{\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \left ( -{\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) +{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \right ) \right ) \ln \left ( F \right ) }{2}{\frac{1}{\sqrt{-\ln \left ( F \right ) bf}}}} \right ){\frac{1}{\sqrt{-\ln \left ( F \right ) bf}}}} \]

3.3.1.2 Problem number 606

\[ \int \frac{F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{d g+e g x} \, dx \] Optimal antiderivative

\[ \frac{\sqrt{\pi } \text{Erfi}\left (a \sqrt{f} \sqrt{\log (F)}+b \sqrt{f} \sqrt{\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 b e \sqrt{f} g n \sqrt{\log (F)}} \] command

int(F^(f*(a+b*ln(c*(e*x+d)^n))^2)/(e*g*x+d*g),x)

Maple 2020 output

\[ \int{\frac{{F}^{f \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}{egx+dg}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{\sqrt{\pi }}{2\,negb}{\it Erf} \left ( -b\sqrt{-f\ln \left ( F \right ) }\ln \left ( \left ( ex+d \right ) ^{n} \right ) +{f \left ( a+b \left ( \ln \left ( c \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \left ( -{\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) +{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \right ) \right ) \right ) \ln \left ( F \right ){\frac{1}{\sqrt{-f\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-f\ln \left ( F \right ) }}}} \]

3.3.2 Test number 63

Test folder name

test_cases/3_Logarithms/3.4_u-a+b_log-c-d+e_x^m-^n-^p

3.3.2.1 Problem number 68

\[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx \] Optimal antiderivative

\[ -\frac{(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{e p x^n \log (x) (f x)^{-n}}{d f}-\frac{e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \] command

int((f*x)^(-1-n)*ln(c*(d+e*x^n)^p),x)

Maple 2020 output

\[ \int \left ( fx \right ) ^{-1-n}\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \, dx \] Maple 2019.2.1 output

\[ -{\frac{x\ln \left ( \left ( d+e{x}^{n} \right ) ^{p} \right ) }{n}{{\rm e}^{-{\frac{ \left ( 1+n \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( ifx \right ) +2\,\ln \left ( f \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}}+{\frac{pe\ln \left ({x}^{n} \right ) }{dn}{{\rm e}^{-{\frac{ \left ( 1+n \right ) \left ( -i\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+2\,\ln \left ( f \right ) \right ) }{2}}}}}-{\frac{pe\ln \left ( d+e{x}^{n} \right ) }{dn}{{\rm e}^{-{\frac{ \left ( 1+n \right ) \left ( -i\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+2\,\ln \left ( f \right ) \right ) }{2}}}}}-{\frac{ \left ( -i\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) \right ) x}{2\,n}{{\rm e}^{-{\frac{ \left ( 1+n \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( ifx \right ) +2\,\ln \left ( f \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \]

3.3.3 Test number 135

Test folder name

test_cases/4_Trig_functions/4.7_Miscellaneous/4.7.1-c_trig-^m-d_trig-^n

3.3.3.1 Problem number 177

\[ \int \cos ^3(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx \] Optimal antiderivative

\[ \frac{7 \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{48 b}+\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \cos (a+b x)}{12 b}-\frac{7 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{64 b}-\frac{7 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{32 b}+\frac{7 \log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{64 b} \] command

int(cos(b*x+a)^3*sin(2*b*x+2*a)^(3/2),x)

Maple 2020 output

\[ \int \left ( \cos \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{{\frac{3}{2}}}\, dx \] Maple 2019.2.1 output

\[ \text{output too large to display} \]

3.3.4 Test number 178

Test folder name

test_cases/6_Hyperbolic_functions/6.5_Hyperbolic_secant/6.5.2-e_x-^m-a+b_sech-c+d_x^n-^p

3.3.4.1 Problem number 77

\[ \int (e x)^{-1+2 n} \left (a+b \text{sech}\left (c+d x^n\right )\right )^2 \, dx \] Optimal antiderivative

\[ -\frac{2 i a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac{2 i a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac{a^2 (e x)^{2 n}}{2 e n}+\frac{4 a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (\cosh \left (c+d x^n\right )\right )}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{2 n} \tanh \left (c+d x^n\right )}{d e n} \] command

int((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n))^2,x)

Maple 2020 output

\[ \int \left ( ex \right ) ^{-1+2\,n} \left ( a+b{\rm sech} \left (c+d{x}^{n}\right ) \right ) ^{2}\, dx \] Maple 2019.2.1 output

\[{\frac{{a}^{2}x{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}}{2\,n}}-2\,{\frac{{b}^{2}x{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}}{nd{{\rm e}^{\ln \left ( x \right ) n}} \left ( \left ({{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}} \right ) ^{2}+1 \right ) }}-{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ( 1+{{\rm e}^{2\,c+2\,d{x}^{n}}} \right ) }{{d}^{2}en}}+{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ({{\rm e}^{2\,c+2\,d{x}^{n}}} \right ) }{{d}^{2}en}}-2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{{\rm e}^{c}}\sqrt{-{{\rm e}^{2\,c}}}{x}^{n}\ln \left ( 1+{{\rm e}^{d{x}^{n}}}\sqrt{-{{\rm e}^{2\,c}}} \right ) }{ed{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}}+2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{{\rm e}^{c}}\sqrt{-{{\rm e}^{2\,c}}}{x}^{n}\ln \left ( 1-{{\rm e}^{d{x}^{n}}}\sqrt{-{{\rm e}^{2\,c}}} \right ) }{ed{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}}-2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{{\rm e}^{c}}\sqrt{-{{\rm e}^{2\,c}}}{\it dilog} \left ( 1+{{\rm e}^{d{x}^{n}}}\sqrt{-{{\rm e}^{2\,c}}} \right ) }{{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}}+2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{{\rm e}^{c}}\sqrt{-{{\rm e}^{2\,c}}}{\it dilog} \left ( 1-{{\rm e}^{d{x}^{n}}}\sqrt{-{{\rm e}^{2\,c}}} \right ) }{{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}} \]

3.3.4.2 Problem number 83

\[ \int \frac{(e x)^{-1+2 n}}{\left (a+b \text{sech}\left (c+d x^n\right )\right )^2} \, dx \] Optimal antiderivative

\[ -\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cosh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}-\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}+1\right )}{a^2 d e n \sqrt{b^2-a^2}}+\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}+1\right )}{a^2 d e n \sqrt{b^2-a^2}}-\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \] command

int((e*x)^(-1+2*n)/(a+b*sech(c+d*x^n))^2,x)

Maple 2020 output

\[ \int{\frac{ \left ( ex \right ) ^{-1+2\,n}}{ \left ( a+b{\rm sech} \left (c+d{x}^{n}\right ) \right ) ^{2}}}\, dx \] Maple 2019.2.1 output

\[{\frac{x{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}}{2\,{a}^{2}n}}-2\,{\frac{{b}^{2}{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}x \left ( b{{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}}+a \right ) }{nd \left ({a}^{2}-{b}^{2} \right ){a}^{2}{{\rm e}^{\ln \left ( x \right ) n}} \left ( \left ({{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}} \right ) ^{2}a+2\,b{{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}}+a \right ) }}-2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}-{b}^{2} \right ) e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}\ln \left ({\frac{-{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}-{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}{-{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}} \right ) }+{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}-{b}^{2} \right ){a}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}\ln \left ({1 \left ( -{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}-{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) \left ( -{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}}}+2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}-{b}^{2} \right ) e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}\ln \left ({\frac{{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}{{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}} \right ) }-{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}-{b}^{2} \right ){a}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}\ln \left ({1 \left ({{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) \left ({{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}}}-2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}-{b}^{2} \right ){d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}{\it dilog} \left ({\frac{-{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}-{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}{-{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}} \right ) }+{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}-{b}^{2} \right ){a}^{2}{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}{\it dilog} \left ({1 \left ( -{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}-{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) \left ( -{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}}}+2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}-{b}^{2} \right ){d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}{\it dilog} \left ({\frac{{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}{{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}} \right ) }-{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}-{b}^{2} \right ){a}^{2}{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}{\it dilog} \left ({1 \left ({{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) \left ({{\rm e}^{c}}b+\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}-{a}^{2}{{\rm e}^{2\,c}}}}}}-{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ( a \left ({{\rm e}^{d{x}^{n}}} \right ) ^{2}{{\rm e}^{2\,c}}+2\,b{{\rm e}^{c}}{{\rm e}^{d{x}^{n}}}+a \right ) }{ \left ({a}^{2}-{b}^{2} \right ){a}^{2}{d}^{2}en}}+2\,{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ({{\rm e}^{d{x}^{n}}} \right ) }{ \left ({a}^{2}-{b}^{2} \right ){a}^{2}{d}^{2}en}} \]

3.3.5 Test number 182

Test folder name

test_cases/6_Hyperbolic_functions/6.6_Hyperbolic_cosecant/6.6.2-e_x-^m-a+b_csch-c+d_x^n-^p

3.3.5.1 Problem number 76

\[ \int (e x)^{-1+2 n} \left (a+b \text{csch}\left (c+d x^n\right )\right )^2 \, dx \] Optimal antiderivative

\[ -\frac{2 a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{2 a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{a^2 (e x)^{2 n}}{2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (\sinh \left (c+d x^n\right )\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{2 n} \coth \left (c+d x^n\right )}{d e n} \] command

int((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n))^2,x)

Maple 2020 output

\[ \int \left ( ex \right ) ^{-1+2\,n} \left ( a+b{\rm csch} \left (c+d{x}^{n}\right ) \right ) ^{2}\, dx \] Maple 2019.2.1 output

\[{\frac{{a}^{2}x{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}}{2\,n}}-2\,{\frac{{b}^{2}x{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}}{nd{{\rm e}^{\ln \left ( x \right ) n}} \left ( \left ({{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}} \right ) ^{2}-1 \right ) }}-{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ({{\rm e}^{2\,c+2\,d{x}^{n}}} \right ) }{{d}^{2}en}}+{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ({{\rm e}^{2\,c+2\,d{x}^{n}}}-1 \right ) }{{d}^{2}en}}+2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}} \left ({{\rm e}^{c}} \right ) ^{2}{x}^{n}\ln \left ( 1-{{\rm e}^{c}}{{\rm e}^{d{x}^{n}}} \right ) }{ed{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}}-2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}} \left ({{\rm e}^{c}} \right ) ^{2}{x}^{n}\ln \left ({{\rm e}^{c}}{{\rm e}^{d{x}^{n}}}+1 \right ) }{ed{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}}+2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}} \left ({{\rm e}^{c}} \right ) ^{2}{\it dilog} \left ( 1-{{\rm e}^{c}}{{\rm e}^{d{x}^{n}}} \right ) }{{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}}-2\,{\frac{ab{e}^{2\,n}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}} \left ({{\rm e}^{c}} \right ) ^{2}{\it dilog} \left ({{\rm e}^{c}}{{\rm e}^{d{x}^{n}}}+1 \right ) }{{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n{{\rm e}^{2\,c}}}} \]

3.3.5.2 Problem number 82

\[ \int \frac{(e x)^{-1+2 n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx \] Optimal antiderivative

\[ -\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 d^2 e n \sqrt{a^2+b^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a^2 d^2 e n \sqrt{a^2+b^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sinh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2+b^2\right )}-\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a^2 d e n \sqrt{a^2+b^2}}+\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}+\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a^2 d e n \sqrt{a^2+b^2}}-\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \] command

int((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x)

Maple 2020 output

\[ \int{\frac{ \left ( ex \right ) ^{-1+2\,n}}{ \left ( a+b{\rm csch} \left (c+d{x}^{n}\right ) \right ) ^{2}}}\, dx \] Maple 2019.2.1 output

\[{\frac{x{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}}{2\,{a}^{2}n}}-2\,{\frac{{b}^{2}{{\rm e}^{ \left ( -1+2\,n \right ) \left ( \ln \left ( e \right ) +\ln \left ( x \right ) -i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) }}x \left ( -b{{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}}+a \right ) }{nd \left ({a}^{2}+{b}^{2} \right ){a}^{2}{{\rm e}^{\ln \left ( x \right ) n}} \left ( \left ({{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}} \right ) ^{2}a+2\,b{{\rm e}^{c+d{{\rm e}^{\ln \left ( x \right ) n}}}}-a \right ) }}-2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}+{b}^{2} \right ) e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}\ln \left ({\frac{{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}{{{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}+{b}^{2} \right ){a}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}\ln \left ({1 \left ({{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) \left ({{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}}}+2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}+{b}^{2} \right ) e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}\ln \left ({\frac{{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}{{{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}} \right ) }+{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{x}^{n}}{d \left ({a}^{2}+{b}^{2} \right ){a}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}\ln \left ({1 \left ({{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) \left ({{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}}}-2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}+{b}^{2} \right ){d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}{\it dilog} \left ({\frac{{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}{{{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}+{b}^{2} \right ){a}^{2}{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}{\it dilog} \left ({1 \left ({{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) \left ({{\rm e}^{c}}b-\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}}}+2\,{\frac{b{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}+{b}^{2} \right ){d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}{\it dilog} \left ({\frac{{{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}{{{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}} \right ) }+{\frac{{b}^{3}{e}^{2\,n}{{\rm e}^{c}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}}{ \left ({a}^{2}+{b}^{2} \right ){a}^{2}{d}^{2}e{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}n}{\it dilog} \left ({1 \left ({{\rm e}^{2\,c}}a{{\rm e}^{d{x}^{n}}}+{{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) \left ({{\rm e}^{c}}b+\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}{{\rm e}^{2\,c}}+{b}^{2} \left ({{\rm e}^{c}} \right ) ^{2}}}}}+{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ( a \left ({{\rm e}^{d{x}^{n}}} \right ) ^{2}{{\rm e}^{2\,c}}+2\,b{{\rm e}^{c}}{{\rm e}^{d{x}^{n}}}-a \right ) }{ \left ({a}^{2}+{b}^{2} \right ){a}^{2}{d}^{2}en}}-2\,{\frac{{b}^{2}{e}^{2\,n}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -1+2\,n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}\ln \left ({{\rm e}^{d{x}^{n}}} \right ) }{ \left ({a}^{2}+{b}^{2} \right ){a}^{2}{d}^{2}en}} \]

3.3.6 Test number 194

Test folder name

test_cases/7_Inverse_hyperbolic_functions/7.3_Inverse_hyperbolic_tangent/7.3.4_u-a+b_arctanh-c_x-^p

3.3.6.1 Problem number 528

\[ \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx \] Optimal antiderivative

\[ -\frac{1}{2} b c e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x}-\frac{c e \left (a+b \tanh ^{-1}(c x)\right )^2}{b}+\frac{1}{2} b c \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right ) \] command

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^2,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{2}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{x}}-{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) \left ( ce\ln \left ( -cx-1 \right ) x-ce\ln \left ( -cx+1 \right ) x+d \right ) }{x}} \]

3.3.6.2 Problem number 529

\[ \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^3} \, dx \] Optimal antiderivative

\[ \frac{1}{2} b c^2 e \text{PolyLog}(2,-c x)-\frac{1}{2} b c^2 e \text{PolyLog}(2,c x)-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x^2}+\frac{1}{2} c^2 e (a+b) \log (1-c x)+\frac{1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right ) \] command

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^3,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{3}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{2\,{x}^{2}}}+{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) \left ( e{c}^{2}\ln \left ( -{c}^{2}{x}^{2}+1 \right ){x}^{2}-2\,e{c}^{2}\ln \left ( x \right ){x}^{2}-d \right ) }{2\,{x}^{2}}} \]

3.3.6.3 Problem number 530

\[ \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^4} \, dx \] Optimal antiderivative

\[ -\frac{1}{6} b c^3 e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac{c^3 e \left (a+b \tanh ^{-1}(c x)\right )^2}{3 b}+\frac{2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{3 x}+\frac{1}{6} b c^3 \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{b c \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 x^2}+\frac{1}{3} b c^3 e \log \left (1-c^2 x^2\right )-b c^3 e \log (x) \] command

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^4,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{4}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{3\,{x}^{3}}}-{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) \left ({c}^{3}e\ln \left ( -cx-1 \right ){x}^{3}-{c}^{3}e\ln \left ( -cx+1 \right ){x}^{3}-2\,e{c}^{2}{x}^{2}+d \right ) }{3\,{x}^{3}}} \]

3.3.6.4 Problem number 531

\[ \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx \] Optimal antiderivative

\[ \frac{1}{4} b c^4 e \text{PolyLog}(2,-c x)-\frac{1}{4} b c^4 e \text{PolyLog}(2,c x)-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}+\frac{1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac{1}{12} c^4 e (3 a-4 b) \log (c x+1)+\frac{a c^2 e}{4 x^2}-\frac{1}{2} a c^4 e \log (x)-\frac{b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}+\frac{5 b c^3 e}{12 x}-\frac{1}{4} b c^4 e \tanh ^{-1}(c x) \] command

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^5,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{5}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{4\,{x}^{4}}}-{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) \left ( 2\,{c}^{4}e\ln \left ( x \right ){x}^{4}-{c}^{4}e\ln \left ( -{c}^{2}{x}^{2}+1 \right ){x}^{4}-e{c}^{2}{x}^{2}+d \right ) }{4\,{x}^{4}}} \]

3.3.6.5 Problem number 532

\[ \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx \] Optimal antiderivative

\[ -\frac{1}{10} b c^5 e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{5 x^5}+\frac{2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}-\frac{c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}+\frac{2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}+\frac{1}{10} b c^5 \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{b c^3 \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 x^2}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 x^4}+\frac{7 b c^3 e}{60 x^2}+\frac{19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac{5}{6} b c^5 e \log (x) \] command

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x^6,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{6}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{5\,{x}^{5}}}-{\frac{ \left ( a-{\frac{i}{2}}b\pi \right ) \left ( 3\,{c}^{5}e\ln \left ( -cx-1 \right ){x}^{5}-3\,{c}^{5}e\ln \left ( -cx+1 \right ){x}^{5}-6\,{c}^{4}e{x}^{4}-2\,e{c}^{2}{x}^{2}+3\,d \right ) }{15\,{x}^{5}}} \]

3.3.7 Test number 198

Test folder name

test_cases/7_Inverse_hyperbolic_functions/7.4_Inverse_hyperbolic_cotangent/7.4.1_Inverse_hyperbolic_cotangent_functions

3.3.7.1 Problem number 172

\[ \int \frac{1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx \] Optimal antiderivative

\[ -\frac{1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac{\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}-\frac{\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \] command

int(1/x/arccoth(tanh(b*x+a))^2,x)

Maple 2020 output

\[ \int{\frac{1}{x \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{2}}}\, dx \] Maple 2019.2.1 output

\[{\frac{4\,i}{bx} \left ( 2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}-\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ){\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ){\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) +\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}-2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}-\pi \, \left ({\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) +2\,\pi \,{\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{2}-\pi \, \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{3}+\pi \,{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}-\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}-2\,\pi -4\,i\ln \left ({{\rm e}^{bx+a}} \right ) \right ) ^{-1}}-16\,{\ln \left ( x \right ) \left ( 2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}+\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ){\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ){\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) -\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{3}-2\,\pi \,{\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{2}-\pi \,{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) +\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}-4\,ibx-2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+4\,i\ln \left ({{\rm e}^{bx+a}} \right ) +2\,\pi \right ) ^{-2}}+{\frac{4\,i}{bx} \left ( 2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}+\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ){\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ){\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) -\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{3}-2\,\pi \,{\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{2}-\pi \,{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) +\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}-4\,ibx-2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+4\,i\ln \left ({{\rm e}^{bx+a}} \right ) +2\,\pi \right ) ^{-1}}+16\,{1\ln \left ( 2\,i\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ){\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ){\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) -i\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+i\pi \, \left ({\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) -2\,i\pi \,{\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{2}+i\pi \, \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{3}-i\pi \,{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+i\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}-2\,i\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}-4\,\ln \left ({{\rm e}^{bx+a}} \right ) +2\,i\pi \right ) \left ( 2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}+\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ){\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ){\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) -\pi \,{\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{3}-2\,\pi \,{\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \left ({\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \right ) ^{2}-\pi \,{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ({{\rm e}^{bx+a}} \right ) ^{2} \right ) +\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{bx+a}} \right ) ^{2}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{3}-4\,ibx-2\,\pi \, \left ({\it csgn} \left ({\frac{i}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}} \right ) \right ) ^{2}+4\,i\ln \left ({{\rm e}^{bx+a}} \right ) +2\,\pi \right ) ^{-2}} \]

3.3.7.2 Problem number 173

\[ \int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx \] Optimal antiderivative

\[ -\frac{2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac{2 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac{2 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \] command

int(1/x^2/arccoth(tanh(b*x+a))^2,x)

Maple 2020 output

\[ \int{\frac{1}{{x}^{2} \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{2}}}\, dx \] Maple 2019.2.1 output

\[ \text{output too large to display} \]

3.3.7.3 Problem number 181

\[ \int \frac{1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx \] Optimal antiderivative

\[ \frac{1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac{1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac{\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac{\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \] command

int(1/x/arccoth(tanh(b*x+a))^3,x)

Maple 2020 output

\[ \int{\frac{1}{x \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{3}}}\, dx \] Maple 2019.2.1 output

\[ \text{output too large to display} \]

3.3.7.4 Problem number 182

\[ \int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx \] Optimal antiderivative

\[ \frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac{3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac{3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \] command

int(1/x^2/arccoth(tanh(b*x+a))^3,x)

Maple 2020 output

\[ \int{\frac{1}{{x}^{2} \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{3}}}\, dx \] Maple 2019.2.1 output

\[ \text{output too large to display} \]

3.3.7.5 Problem number 270

\[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^3} \, dx \] Optimal antiderivative

\[ \frac{1}{2} b c^2 e \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{1}{2} b c^2 e \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x^2}+\frac{1}{2} c^2 e (a+b) \log (1-c x)+\frac{1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x)^2-\frac{1}{2} b c^2 e \coth ^{-1}(c x)^2+b c^2 e \log \left (\frac{2}{1-c x}\right ) \tanh ^{-1}(c x)-b c^2 e \log \left (2-\frac{2}{c x+1}\right ) \coth ^{-1}(c x) \] command

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))/x^3,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{3}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{\ln \left ( -{c}^{2}{x}^{2}+1 \right ) ae}{2\,{x}^{2}}}-{\frac{a \left ( 2\,e{c}^{2}\ln \left ( x \right ){x}^{2}-e{c}^{2}\ln \left ( -{c}^{2}{x}^{2}+1 \right ){x}^{2}+d \right ) }{2\,{x}^{2}}} \]

3.3.7.6 Problem number 271

\[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^5} \, dx \] Optimal antiderivative

\[ \frac{1}{4} b c^4 e \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )+\frac{1}{4} b c^4 e \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}+\frac{1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac{1}{12} c^4 e (3 a-4 b) \log (c x+1)+\frac{a c^2 e}{4 x^2}-\frac{1}{2} a c^4 e \log (x)-\frac{b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac{b c^2 e \coth ^{-1}(c x)}{4 x^2}+\frac{5 b c^3 e}{12 x}-\frac{1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac{1}{4} b c^4 e \tanh ^{-1}(c x)-\frac{1}{4} b c^4 e \coth ^{-1}(c x)^2+\frac{1}{2} b c^4 e \log \left (\frac{2}{1-c x}\right ) \tanh ^{-1}(c x)-\frac{1}{2} b c^4 e \log \left (2-\frac{2}{c x+1}\right ) \coth ^{-1}(c x) \] command

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))/x^5,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{5}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{\ln \left ( -{c}^{2}{x}^{2}+1 \right ) ae}{4\,{x}^{4}}}-{\frac{a \left ( 2\,{c}^{4}e\ln \left ( x \right ){x}^{4}-{c}^{4}e\ln \left ( -{c}^{2}{x}^{2}+1 \right ){x}^{4}-e{c}^{2}{x}^{2}+d \right ) }{4\,{x}^{4}}} \]

3.3.7.7 Problem number 275

\[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx \] Optimal antiderivative

\[ -\frac{1}{2} b c e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x}-\frac{c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}+\frac{1}{2} b c \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right ) \] command

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))/x^2,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{2}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{\ln \left ( -{c}^{2}{x}^{2}+1 \right ) ae}{x}}+{\frac{a \left ( ce\ln \left ( -cx+1 \right ) x-ce\ln \left ( -cx-1 \right ) x-d \right ) }{x}} \]

3.3.7.8 Problem number 276

\[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^4} \, dx \] Optimal antiderivative

\[ -\frac{1}{6} b c^3 e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac{c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}+\frac{1}{6} b c^3 \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{b c \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 x^2}+\frac{1}{3} b c^3 e \log \left (1-c^2 x^2\right )-b c^3 e \log (x) \] command

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))/x^4,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{4}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{\ln \left ( -{c}^{2}{x}^{2}+1 \right ) ae}{3\,{x}^{3}}}-{\frac{a \left ({c}^{3}e\ln \left ( -cx-1 \right ){x}^{3}-{c}^{3}e\ln \left ( -cx+1 \right ){x}^{3}-2\,e{c}^{2}{x}^{2}+d \right ) }{3\,{x}^{3}}} \]

3.3.7.9 Problem number 277

\[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx \] Optimal antiderivative

\[ -\frac{1}{10} b c^5 e \text{PolyLog}\left (2,\frac{1}{1-c^2 x^2}\right )-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{5 x^5}+\frac{2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}-\frac{c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}+\frac{2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}+\frac{1}{10} b c^5 \log \left (1-\frac{1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{b c^3 \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 x^2}-\frac{b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 x^4}+\frac{7 b c^3 e}{60 x^2}+\frac{19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac{5}{6} b c^5 e \log (x) \] command

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))/x^6,x)

Maple 2020 output

\[ \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( -{c}^{2}{x}^{2}+1 \right ) \right ) }{{x}^{6}}}\, dx \] Maple 2019.2.1 output

\[ -{\frac{\ln \left ( -{c}^{2}{x}^{2}+1 \right ) ae}{5\,{x}^{5}}}+{\frac{a \left ( 3\,{c}^{5}e\ln \left ( -cx+1 \right ){x}^{5}-3\,{c}^{5}e\ln \left ( -cx-1 \right ){x}^{5}+6\,{c}^{4}e{x}^{4}+2\,e{c}^{2}{x}^{2}-3\,d \right ) }{15\,{x}^{5}}} \]