Maple 2020 and Maple 2019.2.1

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

$\int \frac{F^{f (a + b \log^{2} (c (d + e x)^{n}))}}{d g + e g x} d x$ Optimal antiderivative

$\frac{\sqrt{π} F^{a f} Erfi (\sqrt{b} \sqrt{f} \sqrt{\log (F)} \log (c (d + e x)^{n}))}{2 \sqrt{b} e \sqrt{f} g n \sqrt{\log (F)}}$ command

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

Maple 2020 output

$\int \frac{F^{f (a + b {(\ln (c {(e x + d)}^{n}))}^{2})}}{e g x + d g} d x$ Maple 2019.2.1 output

$\frac{\sqrt{π} F^{a f}}{2 n e g} E r f (\sqrt{- \ln (F) b f} \ln ({(e x + d)}^{n}) - \frac{b f (2 \ln (c) - i π c s g n (i c {(e x + d)}^{n}) (- c s g n (i c {(e x + d)}^{n}) + c s g n (i c)) (- c s g n (i c {(e x + d)}^{n}) + c s g n (i {(e x + d)}^{n}))) \ln (F)}{2} \frac{1}{\sqrt{- \ln (F) b f}}) \frac{1}{\sqrt{- \ln (F) b f}}$

3.3.1.2 Problem number 606

$\int \frac{F^{f {(a + b \log (c (d + e x)^{n}))}^{2}}}{d g + e g x} d x$ Optimal antiderivative

$\frac{\sqrt{π} Erfi (a \sqrt{f} \sqrt{\log (F)} + b \sqrt{f} \sqrt{\log (F)} \log (c (d + e x)^{n}))}{2 b e \sqrt{f} g n \sqrt{\log (F)}}$ command

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

Maple 2020 output

$\int \frac{F^{f {(a + b \ln (c {(e x + d)}^{n}))}^{2}}}{e g x + d g} d x$ Maple 2019.2.1 output

$- \frac{\sqrt{π}}{2 n e g b} E r f (- b \sqrt{- f \ln (F)} \ln ({(e x + d)}^{n}) + f (a + b (\ln (c) - \frac{i}{2} π c s g n (i c {(e x + d)}^{n}) (- c s g n (i c {(e x + d)}^{n}) + c s g n (i c)) (- c s g n (i c {(e x + d)}^{n}) + c s g n (i {(e x + d)}^{n})))) \ln (F) \frac{1}{\sqrt{- f \ln (F)}}) \frac{1}{\sqrt{- f \ln (F)}}$

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

$\int (f x)^{- 1 - n} \log (c {(d + e x^{n})}^{p}) d x$ Optimal antiderivative

$- \frac{(f x)^{- n} \log (c {(d + e x^{n})}^{p})}{f n} + \frac{e p x^{n} \log (x) (f x)^{- n}}{d f} - \frac{e p x^{n} (f x)^{- n} \log (d + e x^{n})}{d f n}$ command

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

Maple 2020 output

$\int {(f x)}^{- 1 - n} \ln (c {(d + e x^{n})}^{p}) d x$ Maple 2019.2.1 output

$- \frac{x \ln ({(d + e x^{n})}^{p})}{n} e^{- \frac{(1 + n) (- i π {(c s g n (i f x))}^{3} + i π {(c s g n (i f x))}^{2} c s g n (i f) + i π {(c s g n (i f x))}^{2} c s g n (i x) - i π c s g n (i f) c s g n (i x) c s g n (i f x) + 2 \ln (f) + 2 \ln (x))}{2}} + \frac{p e \ln (x^{n})}{d n} e^{- \frac{(1 + n) (- i π c s g n (i f x) c s g n (i f) c s g n (i x) + i π {(c s g n (i f x))}^{2} c s g n (i f) + i π {(c s g n (i f x))}^{2} c s g n (i x) - i π {(c s g n (i f x))}^{3} + 2 \ln (f))}{2}} - \frac{p e \ln (d + e x^{n})}{d n} e^{- \frac{(1 + n) (- i π c s g n (i f x) c s g n (i f) c s g n (i x) + i π {(c s g n (i f x))}^{2} c s g n (i f) + i π {(c s g n (i f x))}^{2} c s g n (i x) - i π {(c s g n (i f x))}^{3} + 2 \ln (f))}{2}} - \frac{(- i π c s g n (i c) c s g n (i {(d + e x^{n})}^{p}) c s g n (i c {(d + e x^{n})}^{p}) + i π c s g n (i c) {(c s g n (i c {(d + e x^{n})}^{p}))}^{2} + i π c s g n (i {(d + e x^{n})}^{p}) {(c s g n (i c {(d + e x^{n})}^{p}))}^{2} - i π {(c s g n (i c {(d + e x^{n})}^{p}))}^{3} + 2 \ln (c)) x}{2 n} e^{- \frac{(1 + n) (- i π {(c s g n (i f x))}^{3} + i π {(c s g n (i f x))}^{2} c s g n (i f) + i π {(c s g n (i f x))}^{2} c s g n (i x) - i π c s g n (i f) c s g n (i x) c s g n (i f x) + 2 \ln (f) + 2 \ln (x))}{2}}$

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

$\int \cos^{3} (a + b x) \sin^{\frac{3}{2}} (2 a + 2 b x) d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int {(\cos (b x + a))}^{3} {(\sin (2 b x + 2 a))}^{\frac{3}{2}} d x$ Maple 2019.2.1 output

$output too large to display$

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

$\int (e x)^{- 1 + 2 n} {(a + b sech (c + d x^{n}))}^{2} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int {(e x)}^{- 1 + 2 n} {(a + b s e c h (c + d x^{n}))}^{2} d x$ Maple 2019.2.1 output

$\frac{a^{2} x e^{(- 1 + 2 n) (\ln (e) + \ln (x) - \frac{i}{2} π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))}}{2 n} - 2 \frac{b^{2} x e^{(- 1 + 2 n) (\ln (e) + \ln (x) - i / 2 π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))}}{n d e^{\ln (x) n} ({(e^{c + d e^{\ln (x) n}})}^{2} + 1)} - \frac{b^{2} e^{2 n} e^{- \frac{i}{2} π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (1 + e^{2 c + 2 d x^{n}})}{d^{2} e n} + \frac{b^{2} e^{2 n} e^{- \frac{i}{2} π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (e^{2 c + 2 d x^{n}})}{d^{2} e n} - 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} e^{c} \sqrt{- e^{2 c}} x^{n} \ln (1 + e^{d x^{n}} \sqrt{- e^{2 c}})}{e d e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}} + 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} e^{c} \sqrt{- e^{2 c}} x^{n} \ln (1 - e^{d x^{n}} \sqrt{- e^{2 c}})}{e d e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}} - 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} e^{c} \sqrt{- e^{2 c}} d i l o g (1 + e^{d x^{n}} \sqrt{- e^{2 c}})}{d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}} + 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} e^{c} \sqrt{- e^{2 c}} d i l o g (1 - e^{d x^{n}} \sqrt{- e^{2 c}})}{d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}}$

3.3.4.2 Problem number 83

$\int \frac{(e x)^{- 1 + 2 n}}{{(a + b sech (c + d x^{n}))}^{2}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{{(e x)}^{- 1 + 2 n}}{{(a + b s e c h (c + d x^{n}))}^{2}} d x$ Maple 2019.2.1 output

$\frac{x e^{(- 1 + 2 n) (\ln (e) + \ln (x) - \frac{i}{2} π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))}}{2 a^{2} n} - 2 \frac{b^{2} e^{(- 1 + 2 n) (\ln (e) + \ln (x) - i / 2 π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))} x (b e^{c + d e^{\ln (x) n}} + a)}{n d (a^{2} - b^{2}) a^{2} e^{\ln (x) n} ({(e^{c + d e^{\ln (x) n}})}^{2} a + 2 b e^{c + d e^{\ln (x) n}} + a)} - 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} - b^{2}) e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} \ln (\frac{- e^{2 c} a e^{d x^{n}} - e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}{- e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}) + \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} - b^{2}) a^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} \ln (1 (- e^{2 c} a e^{d x^{n}} - e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}) {(- e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}})}^{- 1}) \frac{1}{\sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} + 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} - b^{2}) e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} \ln (\frac{e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}{e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}) - \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} - b^{2}) a^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} \ln (1 (e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}) {(e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}})}^{- 1}) \frac{1}{\sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} - 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}}}{(a^{2} - b^{2}) d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} d i l o g (\frac{- e^{2 c} a e^{d x^{n}} - e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}{- e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}) + \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}}}{(a^{2} - b^{2}) a^{2} d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} d i l o g (1 (- e^{2 c} a e^{d x^{n}} - e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}) {(- e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}})}^{- 1}) \frac{1}{\sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} + 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}}}{(a^{2} - b^{2}) d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} d i l o g (\frac{e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}{e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}}) - \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}}}{(a^{2} - b^{2}) a^{2} d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} d i l o g (1 (e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}) {(e^{c} b + \sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}})}^{- 1}) \frac{1}{\sqrt{b^{2} {(e^{c})}^{2} - a^{2} e^{2 c}}} - \frac{b^{2} e^{2 n} e^{- \frac{i}{2} π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (a {(e^{d x^{n}})}^{2} e^{2 c} + 2 b e^{c} e^{d x^{n}} + a)}{(a^{2} - b^{2}) a^{2} d^{2} e n} + 2 \frac{b^{2} e^{2 n} e^{- i / 2 π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (e^{d x^{n}})}{(a^{2} - b^{2}) a^{2} d^{2} e n}$

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

$\int (e x)^{- 1 + 2 n} {(a + b csch (c + d x^{n}))}^{2} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int {(e x)}^{- 1 + 2 n} {(a + b c s c h (c + d x^{n}))}^{2} d x$ Maple 2019.2.1 output

$\frac{a^{2} x e^{(- 1 + 2 n) (\ln (e) + \ln (x) - \frac{i}{2} π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))}}{2 n} - 2 \frac{b^{2} x e^{(- 1 + 2 n) (\ln (e) + \ln (x) - i / 2 π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))}}{n d e^{\ln (x) n} ({(e^{c + d e^{\ln (x) n}})}^{2} - 1)} - \frac{b^{2} e^{2 n} e^{- \frac{i}{2} π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (e^{2 c + 2 d x^{n}})}{d^{2} e n} + \frac{b^{2} e^{2 n} e^{- \frac{i}{2} π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (e^{2 c + 2 d x^{n}} - 1)}{d^{2} e n} + 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} {(e^{c})}^{2} x^{n} \ln (1 - e^{c} e^{d x^{n}})}{e d e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}} - 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} {(e^{c})}^{2} x^{n} \ln (e^{c} e^{d x^{n}} + 1)}{e d e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}} + 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} {(e^{c})}^{2} d i l o g (1 - e^{c} e^{d x^{n}})}{d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}} - 2 \frac{a b e^{2 n} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} {(e^{c})}^{2} d i l o g (e^{c} e^{d x^{n}} + 1)}{d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n e^{2 c}}$

3.3.5.2 Problem number 82

$\int \frac{(e x)^{- 1 + 2 n}}{{(a + b csch (c + d x^{n}))}^{2}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{{(e x)}^{- 1 + 2 n}}{{(a + b c s c h (c + d x^{n}))}^{2}} d x$ Maple 2019.2.1 output

$\frac{x e^{(- 1 + 2 n) (\ln (e) + \ln (x) - \frac{i}{2} π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))}}{2 a^{2} n} - 2 \frac{b^{2} e^{(- 1 + 2 n) (\ln (e) + \ln (x) - i / 2 π c s g n (i e x) (- c s g n (i e x) + c s g n (i e)) (- c s g n (i e x) + c s g n (i x)))} x (- b e^{c + d e^{\ln (x) n}} + a)}{n d (a^{2} + b^{2}) a^{2} e^{\ln (x) n} ({(e^{c + d e^{\ln (x) n}})}^{2} a + 2 b e^{c + d e^{\ln (x) n}} - a)} - 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} + b^{2}) e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} \ln (\frac{e^{2 c} a e^{d x^{n}} + e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}{e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}) - \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} + b^{2}) a^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} \ln (1 (e^{2 c} a e^{d x^{n}} + e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}) {(e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}})}^{- 1}) \frac{1}{\sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} + 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} + b^{2}) e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} \ln (\frac{e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}{e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}) + \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}} x^{n}}{d (a^{2} + b^{2}) a^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} \ln (1 (e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}) {(e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}})}^{- 1}) \frac{1}{\sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} - 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}}}{(a^{2} + b^{2}) d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} d i l o g (\frac{e^{2 c} a e^{d x^{n}} + e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}{e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}) - \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}}}{(a^{2} + b^{2}) a^{2} d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} d i l o g (1 (e^{2 c} a e^{d x^{n}} + e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}) {(e^{c} b - \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}})}^{- 1}) \frac{1}{\sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} + 2 \frac{b e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- i / 2 π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- i / 2 π c s g n (i x) {(c s g n (i e x))}^{2}} e^{i / 2 π {(c s g n (i e x))}^{3}}}{(a^{2} + b^{2}) d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} d i l o g (\frac{e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}{e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}}) + \frac{b^{3} e^{2 n} e^{c} e^{i π n c s g n (i e) {(c s g n (i e x))}^{2}} e^{i π n c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π c s g n (i e) c s g n (i x) c s g n (i e x)} e^{- \frac{i}{2} π c s g n (i e) {(c s g n (i e x))}^{2}} e^{- \frac{i}{2} π c s g n (i x) {(c s g n (i e x))}^{2}} e^{\frac{i}{2} π {(c s g n (i e x))}^{3}}}{(a^{2} + b^{2}) a^{2} d^{2} e e^{i π n c s g n (i e) c s g n (i x) c s g n (i e x)} e^{i π n {(c s g n (i e x))}^{3}} n} d i l o g (1 (e^{2 c} a e^{d x^{n}} + e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}) {(e^{c} b + \sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}})}^{- 1}) \frac{1}{\sqrt{a^{2} e^{2 c} + b^{2} {(e^{c})}^{2}}} + \frac{b^{2} e^{2 n} e^{- \frac{i}{2} π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (a {(e^{d x^{n}})}^{2} e^{2 c} + 2 b e^{c} e^{d x^{n}} - a)}{(a^{2} + b^{2}) a^{2} d^{2} e n} - 2 \frac{b^{2} e^{2 n} e^{- i / 2 π c s g n (i e x) (- 1 + 2 n) (- c s g n (i e x) + c s g n (i x)) (- c s g n (i e x) + c s g n (i e))} \ln (e^{d x^{n}})}{(a^{2} + b^{2}) a^{2} d^{2} e n}$

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

$\int \frac{(a + b \tanh^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{2}} d x$ Optimal antiderivative

$- \frac{1}{2} b c e PolyLog (2, \frac{1}{1 - c^{2} x^{2}}) - \frac{(a + b \tanh^{- 1} (c x)) (e \log (1 - c^{2} x^{2}) + d)}{x} - \frac{c e {(a + b \tanh^{- 1} (c x))}^{2}}{b} + \frac{1}{2} b c \log (1 - \frac{1}{1 - c^{2} x^{2}}) (e \log (1 - c^{2} x^{2}) + d)$ command

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

Maple 2020 output

$\int \frac{(a + b A r t a n h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{2}} d x$ Maple 2019.2.1 output

$- \frac{(a - \frac{i}{2} b π) e \ln (- c^{2} x^{2} + 1)}{x} - \frac{(a - \frac{i}{2} b π) (c e \ln (- c x - 1) x - c e \ln (- c x + 1) x + d)}{x}$

3.3.6.2 Problem number 529

$\int \frac{(a + b \tanh^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{3}} d x$ Optimal antiderivative

$\frac{1}{2} b c^{2} e PolyLog (2, - c x) - \frac{1}{2} b c^{2} e PolyLog (2, c x) - \frac{(a + b \tanh^{- 1} (c x)) (e \log (1 - c^{2} x^{2}) + d)}{2 x^{2}} + \frac{1}{2} c^{2} e (a + b) \log (1 - c x) + \frac{1}{2} c^{2} e (a - b) \log (c x + 1) - a c^{2} e \log (x) - \frac{b c (e \log (1 - c^{2} x^{2}) + d)}{2 x} + \frac{1}{2} b c^{2} \tanh^{- 1} (c x) (e \log (1 - c^{2} x^{2}) + d)$ command

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

Maple 2020 output

$\int \frac{(a + b A r t a n h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{3}} d x$ Maple 2019.2.1 output

$- \frac{(a - \frac{i}{2} b π) e \ln (- c^{2} x^{2} + 1)}{2 x^{2}} + \frac{(a - \frac{i}{2} b π) (e c^{2} \ln (- c^{2} x^{2} + 1) x^{2} - 2 e c^{2} \ln (x) x^{2} - d)}{2 x^{2}}$

3.3.6.3 Problem number 530

$\int \frac{(a + b \tanh^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{4}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{(a + b A r t a n h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{4}} d x$ Maple 2019.2.1 output

$- \frac{(a - \frac{i}{2} b π) e \ln (- c^{2} x^{2} + 1)}{3 x^{3}} - \frac{(a - \frac{i}{2} b π) (c^{3} e \ln (- c x - 1) x^{3} - c^{3} e \ln (- c x + 1) x^{3} - 2 e c^{2} x^{2} + d)}{3 x^{3}}$

3.3.6.4 Problem number 531

$\int \frac{(a + b \tanh^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{5}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{(a + b A r t a n h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{5}} d x$ Maple 2019.2.1 output

$- \frac{(a - \frac{i}{2} b π) e \ln (- c^{2} x^{2} + 1)}{4 x^{4}} - \frac{(a - \frac{i}{2} b π) (2 c^{4} e \ln (x) x^{4} - c^{4} e \ln (- c^{2} x^{2} + 1) x^{4} - e c^{2} x^{2} + d)}{4 x^{4}}$

3.3.6.5 Problem number 532

$\int \frac{(a + b \tanh^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{6}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{(a + b A r t a n h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{6}} d x$ Maple 2019.2.1 output

$- \frac{(a - \frac{i}{2} b π) e \ln (- c^{2} x^{2} + 1)}{5 x^{5}} - \frac{(a - \frac{i}{2} b π) (3 c^{5} e \ln (- c x - 1) x^{5} - 3 c^{5} e \ln (- c x + 1) x^{5} - 6 c^{4} e x^{4} - 2 e c^{2} x^{2} + 3 d)}{15 x^{5}}$

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

$\int \frac{1}{x \coth^{- 1} (\tanh (a + b x))^{2}} d x$ Optimal antiderivative

$- \frac{1}{(b x - \coth^{- 1} (\tanh (a + b x))) \coth^{- 1} (\tanh (a + b x))} + \frac{\log (x)}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{2}} - \frac{\log (\coth^{- 1} (\tanh (a + b x)))}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{2}}$ command

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

Maple 2020 output

$\int \frac{1}{x {(a r c c o t h (\tanh (b x + a)))}^{2}} d x$ Maple 2019.2.1 output

$\frac{4 i}{b x} {(2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{2} - π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) c s g n (i {(e^{b x + a})}^{2}) c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}) + π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} - 2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{3} - π {(c s g n (i e^{b x + a}))}^{2} c s g n (i {(e^{b x + a})}^{2}) + 2 π c s g n (i e^{b x + a}) {(c s g n (i {(e^{b x + a})}^{2}))}^{2} - π {(c s g n (i {(e^{b x + a})}^{2}))}^{3} + π c s g n (i {(e^{b x + a})}^{2}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} - π {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{3} - 2 π - 4 i \ln (e^{b x + a}))}^{- 1} - 16 \ln (x) {(2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{3} + π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) c s g n (i {(e^{b x + a})}^{2}) c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}) - π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + π {(c s g n (i {(e^{b x + a})}^{2}))}^{3} - 2 π c s g n (i e^{b x + a}) {(c s g n (i {(e^{b x + a})}^{2}))}^{2} - π c s g n (i {(e^{b x + a})}^{2}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + π {(c s g n (i e^{b x + a}))}^{2} c s g n (i {(e^{b x + a})}^{2}) + π {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{3} - 4 i b x - 2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{2} + 4 i \ln (e^{b x + a}) + 2 π)}^{- 2} + \frac{4 i}{b x} {(2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{3} + π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) c s g n (i {(e^{b x + a})}^{2}) c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}) - π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + π {(c s g n (i {(e^{b x + a})}^{2}))}^{3} - 2 π c s g n (i e^{b x + a}) {(c s g n (i {(e^{b x + a})}^{2}))}^{2} - π c s g n (i {(e^{b x + a})}^{2}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + π {(c s g n (i e^{b x + a}))}^{2} c s g n (i {(e^{b x + a})}^{2}) + π {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{3} - 4 i b x - 2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{2} + 4 i \ln (e^{b x + a}) + 2 π)}^{- 1} + 16 1 \ln (2 i π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{3} + i π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) c s g n (i {(e^{b x + a})}^{2}) c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}) - i π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + i π {(c s g n (i e^{b x + a}))}^{2} c s g n (i {(e^{b x + a})}^{2}) - 2 i π c s g n (i e^{b x + a}) {(c s g n (i {(e^{b x + a})}^{2}))}^{2} + i π {(c s g n (i {(e^{b x + a})}^{2}))}^{3} - i π c s g n (i {(e^{b x + a})}^{2}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + i π {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{3} - 2 i π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{2} - 4 \ln (e^{b x + a}) + 2 i π) {(2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{3} + π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) c s g n (i {(e^{b x + a})}^{2}) c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}) - π c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + π {(c s g n (i {(e^{b x + a})}^{2}))}^{3} - 2 π c s g n (i e^{b x + a}) {(c s g n (i {(e^{b x + a})}^{2}))}^{2} - π c s g n (i {(e^{b x + a})}^{2}) {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{2} + π {(c s g n (i e^{b x + a}))}^{2} c s g n (i {(e^{b x + a})}^{2}) + π {(c s g n (\frac{i {(e^{b x + a})}^{2}}{{(e^{b x + a})}^{2} + 1}))}^{3} - 4 i b x - 2 π {(c s g n (\frac{i}{{(e^{b x + a})}^{2} + 1}))}^{2} + 4 i \ln (e^{b x + a}) + 2 π)}^{- 2}$

3.3.7.2 Problem number 173

$\int \frac{1}{x^{2} \coth^{- 1} (\tanh (a + b x))^{2}} d x$ Optimal antiderivative

$- \frac{2 b}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{2} \coth^{- 1} (\tanh (a + b x))} + \frac{1}{x (b x - \coth^{- 1} (\tanh (a + b x))) \coth^{- 1} (\tanh (a + b x))} + \frac{2 b \log (x)}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{3}} - \frac{2 b \log (\coth^{- 1} (\tanh (a + b x)))}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{3}}$ command

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

Maple 2020 output

$\int \frac{1}{x^{2} {(a r c c o t h (\tanh (b x + a)))}^{2}} d x$ Maple 2019.2.1 output

$output too large to display$

3.3.7.3 Problem number 181

$\int \frac{1}{x \coth^{- 1} (\tanh (a + b x))^{3}} d x$ Optimal antiderivative

$\frac{1}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{2} \coth^{- 1} (\tanh (a + b x))} - \frac{1}{2 (b x - \coth^{- 1} (\tanh (a + b x))) \coth^{- 1} (\tanh (a + b x))^{2}} - \frac{\log (x)}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{3}} + \frac{\log (\coth^{- 1} (\tanh (a + b x)))}{{(b x - \coth^{- 1} (\tanh (a + b x)))}^{3}}$ command

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

Maple 2020 output

$\int \frac{1}{x {(a r c c o t h (\tanh (b x + a)))}^{3}} d x$ Maple 2019.2.1 output

$output too large to display$

3.3.7.4 Problem number 182

$\int \frac{1}{x^{2} \coth^{- 1} (\tanh (a + b x))^{3}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{1}{x^{2} {(a r c c o t h (\tanh (b x + a)))}^{3}} d x$ Maple 2019.2.1 output

$output too large to display$

3.3.7.5 Problem number 270

$\int \frac{(a + b \coth^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{3}} d x$ Optimal antiderivative

$\frac{1}{2} b c^{2} e PolyLog (2, 1 - \frac{2}{1 - c x}) + \frac{1}{2} b c^{2} e PolyLog (2, \frac{2}{c x + 1} - 1) - \frac{(a + b \coth^{- 1} (c x)) (e \log (1 - c^{2} x^{2}) + d)}{2 x^{2}} + \frac{1}{2} c^{2} e (a + b) \log (1 - c x) + \frac{1}{2} c^{2} e (a - b) \log (c x + 1) - a c^{2} e \log (x) - \frac{b c (e \log (1 - c^{2} x^{2}) + d)}{2 x} + \frac{1}{2} b c^{2} \tanh^{- 1} (c x) (e \log (1 - c^{2} x^{2}) + d) - \frac{1}{2} b c^{2} e \tanh^{- 1} (c x)^{2} - \frac{1}{2} b c^{2} e \coth^{- 1} (c x)^{2} + b c^{2} e \log (\frac{2}{1 - c x}) \tanh^{- 1} (c x) - b c^{2} e \log (2 - \frac{2}{c x + 1}) \coth^{- 1} (c x)$ command

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

Maple 2020 output

$\int \frac{(a + b a r c c o t h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{3}} d x$ Maple 2019.2.1 output

$- \frac{\ln (- c^{2} x^{2} + 1) a e}{2 x^{2}} - \frac{a (2 e c^{2} \ln (x) x^{2} - e c^{2} \ln (- c^{2} x^{2} + 1) x^{2} + d)}{2 x^{2}}$

3.3.7.6 Problem number 271

$\int \frac{(a + b \coth^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{5}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{(a + b a r c c o t h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{5}} d x$ Maple 2019.2.1 output

$- \frac{\ln (- c^{2} x^{2} + 1) a e}{4 x^{4}} - \frac{a (2 c^{4} e \ln (x) x^{4} - c^{4} e \ln (- c^{2} x^{2} + 1) x^{4} - e c^{2} x^{2} + d)}{4 x^{4}}$

3.3.7.7 Problem number 275

$\int \frac{(a + b \coth^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{2}} d x$ Optimal antiderivative

$- \frac{1}{2} b c e PolyLog (2, \frac{1}{1 - c^{2} x^{2}}) - \frac{(a + b \coth^{- 1} (c x)) (e \log (1 - c^{2} x^{2}) + d)}{x} - \frac{c e {(a + b \coth^{- 1} (c x))}^{2}}{b} + \frac{1}{2} b c \log (1 - \frac{1}{1 - c^{2} x^{2}}) (e \log (1 - c^{2} x^{2}) + d)$ command

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

Maple 2020 output

$\int \frac{(a + b a r c c o t h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{2}} d x$ Maple 2019.2.1 output

$- \frac{\ln (- c^{2} x^{2} + 1) a e}{x} + \frac{a (c e \ln (- c x + 1) x - c e \ln (- c x - 1) x - d)}{x}$

3.3.7.8 Problem number 276

$\int \frac{(a + b \coth^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{4}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{(a + b a r c c o t h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{4}} d x$ Maple 2019.2.1 output

$- \frac{\ln (- c^{2} x^{2} + 1) a e}{3 x^{3}} - \frac{a (c^{3} e \ln (- c x - 1) x^{3} - c^{3} e \ln (- c x + 1) x^{3} - 2 e c^{2} x^{2} + d)}{3 x^{3}}$

3.3.7.9 Problem number 277

$\int \frac{(a + b \coth^{- 1} (c x)) (d + e \log (1 - c^{2} x^{2}))}{x^{6}} d x$ Optimal antiderivative

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

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

Maple 2020 output

$\int \frac{(a + b a r c c o t h (c x)) (d + e \ln (- c^{2} x^{2} + 1))}{x^{6}} d x$ Maple 2019.2.1 output

$- \frac{\ln (- c^{2} x^{2} + 1) a e}{5 x^{5}} + \frac{a (3 c^{5} e \ln (- c x + 1) x^{5} - 3 c^{5} e \ln (- c x - 1) x^{5} + 6 c^{4} e x^{4} + 2 e c^{2} x^{2} - 3 d)}{15 x^{5}}$

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