Optimal. Leaf size=433 \[ -\frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac{2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \sqrt{a^2+b^2}}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d} \]
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Rubi [A] time = 0.814057, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5575, 4182, 2531, 2282, 6589, 3322, 2264, 2190} \[ -\frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac{2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \sqrt{a^2+b^2}}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 5575
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 3322
Rule 2264
Rule 2190
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \text{csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \text{csch}(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{(2 b) \int \frac{e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a}-\frac{(2 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac{(2 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \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)^2}{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)^2}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a \sqrt{a^2+b^2}}+\frac{\left (2 f^2\right ) \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac{\left (2 f^2\right ) \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^2 \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)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{(2 b f) \int (e+f x) \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{(2 b f) \int (e+f x) \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 (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^2 \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)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{2 b f (e+f x) \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{2 b f (e+f x) \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{2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{\left (2 b f^2\right ) \int \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 (2 b f^2\right ) \int \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{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^2 \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)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{2 b f (e+f x) \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{2 b f (e+f x) \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{2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^3}-\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\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^3}\\ &=-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{b (e+f x)^2 \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)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{2 b f (e+f x) \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{2 b f (e+f x) \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{2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac{2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{2 b f^2 \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{2 b f^2 \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}\\ \end{align*}
Mathematica [A] time = 2.12001, size = 454, normalized size = 1.05 \[ \frac{\frac{b \left (-2 d f (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+2 d f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+2 f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+2 d^2 e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-2 d^2 e f x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+2 d^2 e f x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-d^2 f^2 x^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+d^2 f^2 x^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )\right )}{\sqrt{a^2+b^2}}-2 d f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )+2 d f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )+2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )-2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )+d^2 (e+f x)^2 \log \left (1-e^{c+d x}\right )-d^2 (e+f x)^2 \log \left (e^{c+d x}+1\right )}{a d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.334, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}{\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 = 2.68204, size = 2674, normalized size = 6.18 \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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