Optimal. Leaf size=154 \[ \frac{d (c+d x) \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac{d (c+d x) \text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac{d^2 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac{d^2 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}+\frac{(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.165645, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4186, 3770, 4182, 2531, 2282, 6589} \[ \frac{d (c+d x) \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac{d (c+d x) \text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac{d^2 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac{d^2 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}+\frac{(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 4186
Rule 3770
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^2 \text{csch}^3(a+b x) \, dx &=-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b}-\frac{1}{2} \int (c+d x)^2 \text{csch}(a+b x) \, dx+\frac{d^2 \int \text{csch}(a+b x) \, dx}{b^2}\\ &=\frac{(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b}+\frac{d \int (c+d x) \log \left (1-e^{a+b x}\right ) \, dx}{b}-\frac{d \int (c+d x) \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=\frac{(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b}+\frac{d (c+d x) \text{Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac{d (c+d x) \text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{d^2 \int \text{Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac{d^2 \int \text{Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac{(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b}+\frac{d (c+d x) \text{Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac{d (c+d x) \text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=\frac{(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac{d (c+d x) \text{csch}(a+b x)}{b^2}-\frac{(c+d x)^2 \coth (a+b x) \text{csch}(a+b x)}{2 b}+\frac{d (c+d x) \text{Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac{d (c+d x) \text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{d^2 \text{Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac{d^2 \text{Li}_3\left (e^{a+b x}\right )}{b^3}\\ \end{align*}
Mathematica [B] time = 10.7472, size = 420, normalized size = 2.73 \[ \frac{2 b d (c+d x) \text{PolyLog}\left (2,-e^{a+b x}\right )-2 b d (c+d x) \text{PolyLog}\left (2,e^{a+b x}\right )-2 d^2 \text{PolyLog}\left (3,-e^{a+b x}\right )+2 d^2 \text{PolyLog}\left (3,e^{a+b x}\right )+b^2 \left (-c^2\right ) \log \left (1-e^{a+b x}\right )+b^2 c^2 \log \left (e^{a+b x}+1\right )-2 b^2 c d x \log \left (1-e^{a+b x}\right )+2 b^2 c d x \log \left (e^{a+b x}+1\right )-b^2 d^2 x^2 \log \left (1-e^{a+b x}\right )+b^2 d^2 x^2 \log \left (e^{a+b x}+1\right )+2 d^2 \log \left (1-e^{a+b x}\right )-2 d^2 \log \left (e^{a+b x}+1\right )}{2 b^3}+\frac{\text{csch}\left (\frac{a}{2}\right ) \text{csch}\left (\frac{a}{2}+\frac{b x}{2}\right ) \left (c d \sinh \left (\frac{b x}{2}\right )+d^2 x \sinh \left (\frac{b x}{2}\right )\right )}{2 b^2}+\frac{\text{sech}\left (\frac{a}{2}\right ) \text{sech}\left (\frac{a}{2}+\frac{b x}{2}\right ) \left (c d \sinh \left (\frac{b x}{2}\right )+d^2 x \sinh \left (\frac{b x}{2}\right )\right )}{2 b^2}-\frac{d \text{csch}(a) (c+d x)}{b^2}+\frac{\left (-c^2-2 c d x-d^2 x^2\right ) \text{csch}^2\left (\frac{a}{2}+\frac{b x}{2}\right )}{8 b}+\frac{\left (-c^2-2 c d x-d^2 x^2\right ) \text{sech}^2\left (\frac{a}{2}+\frac{b x}{2}\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 444, normalized size = 2.9 \begin{align*} -{\frac{{{\rm e}^{bx+a}} \left ( b{d}^{2}{x}^{2}{{\rm e}^{2\,bx+2\,a}}+2\,bcdx{{\rm e}^{2\,bx+2\,a}}+b{c}^{2}{{\rm e}^{2\,bx+2\,a}}+b{d}^{2}{x}^{2}+2\,{d}^{2}x{{\rm e}^{2\,bx+2\,a}}+2\,bcdx+2\,cd{{\rm e}^{2\,bx+2\,a}}+b{c}^{2}-2\,{d}^{2}x-2\,cd \right ) }{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}-{\frac{{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{{d}^{2}{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-2\,{\frac{{d}^{2}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{cd\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{cd\ln \left ( 1+{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}-{\frac{cd\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{b}}-{\frac{cd\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}-2\,{\frac{cda{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{{d}^{2}\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{2}}{2\,b}}+{\frac{{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) x}{{b}^{2}}}-{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{2}}{2\,b}}-{\frac{{d}^{2}{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) x}{{b}^{2}}}+{\frac{cd{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-{\frac{cd{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{{a}^{2}{d}^{2}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{{c}^{2}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{b}}-{\frac{{d}^{2}\ln \left ( 1+{{\rm e}^{bx+a}} \right ){a}^{2}}{2\,{b}^{3}}}+{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{2}}{2\,{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.84349, size = 531, normalized size = 3.45 \begin{align*} \frac{1}{2} \, c^{2}{\left (\frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac{2 \,{\left (e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}\right )}}{b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}}\right )} + \frac{{\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac{{\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac{{\left (b d^{2} x^{2} e^{\left (3 \, a\right )} + 2 \, c d e^{\left (3 \, a\right )} + 2 \,{\left (b c d + d^{2}\right )} x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} +{\left (b d^{2} x^{2} e^{a} - 2 \, c d e^{a} + 2 \,{\left (b c d - d^{2}\right )} x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac{{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x{\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{2}}{2 \, b^{3}} - \frac{{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x{\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{2}}{2 \, b^{3}} - \frac{d^{2} \log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac{d^{2} \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 3.16852, size = 5252, normalized size = 34.1 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{2} \operatorname{csch}^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \operatorname{csch}\left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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