Optimal. Leaf size=605 \[ \frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \sqrt{a^2+b^2}}-\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \sqrt{a^2+b^2}}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d} \]
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Rubi [A] time = 0.964937, antiderivative size = 605, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5575, 4182, 2531, 6609, 2282, 6589, 3322, 2264, 2190} \[ \frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \sqrt{a^2+b^2}}-\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \sqrt{a^2+b^2}}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 5575
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 3322
Rule 2264
Rule 2190
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \text{csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \text{csch}(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{(2 b) \int \frac{e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a}-\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{\left (2 b^2\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a \sqrt{a^2+b^2}}+\frac{\left (2 b^2\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a \sqrt{a^2+b^2}}+\frac{\left (6 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac{\left (6 f^2\right ) \int (e+f x) \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{(3 b f) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a \sqrt{a^2+b^2} d}-\frac{(3 b f) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a \sqrt{a^2+b^2} d}-\frac{\left (6 f^3\right ) \int \text{Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}+\frac{\left (6 f^3\right ) \int \text{Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{\left (6 b f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a \sqrt{a^2+b^2} d^2}-\frac{\left (6 b f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a \sqrt{a^2+b^2} d^2}-\frac{\left (6 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac{\left (6 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{6 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac{\left (6 b f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a \sqrt{a^2+b^2} d^3}+\frac{\left (6 b f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a \sqrt{a^2+b^2} d^3}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{6 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac{\left (6 b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{\left (6 b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \sqrt{a^2+b^2} d^4}\\ &=-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{6 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac{6 b f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{6 b f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}\\ \end{align*}
Mathematica [A] time = 3.11956, size = 757, normalized size = 1.25 \[ \frac{\frac{b \left (-3 d^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+3 d^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+6 d e f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-6 d e f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+6 d f^3 x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-6 d f^3 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-6 f^3 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-3 d^3 e^2 f x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+3 d^3 e^2 f x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+2 d^3 e^3 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-3 d^3 e f^2 x^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+3 d^3 e f^2 x^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-d^3 f^3 x^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+d^3 f^3 x^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )\right )}{\sqrt{a^2+b^2}}-3 f \left (d^2 (e+f x)^2 \text{PolyLog}(2,-\sinh (c+d x)-\cosh (c+d x))-2 d f (e+f x) \text{PolyLog}(3,-\sinh (c+d x)-\cosh (c+d x))+2 f^2 \text{PolyLog}(4,-\sinh (c+d x)-\cosh (c+d x))\right )+3 f \left (d^2 (e+f x)^2 \text{PolyLog}(2,\sinh (c+d x)+\cosh (c+d x))-2 d f (e+f x) \text{PolyLog}(3,\sinh (c+d x)+\cosh (c+d x))+2 f^2 \text{PolyLog}(4,\sinh (c+d x)+\cosh (c+d x))\right )-2 d^3 (e+f x)^3 \tanh ^{-1}(\sinh (c+d x)+\cosh (c+d x))}{a d^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}{\rm csch} \left (dx+c\right )}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 3.05163, size = 3903, normalized size = 6.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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