Optimal. Leaf size=99 \[ \frac {2 d^2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.09, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4182, 2531, 2282, 6589} \[ -\frac {2 d (c+d x) \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 d^2 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^2 \text {csch}(a+b x) \, dx &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(2 d) \int (c+d x) \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {(2 d) \int (c+d x) \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\left (2 d^2\right ) \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (2 d^2\right ) \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 d^2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 1.96, size = 118, normalized size = 1.19 \[ \frac {-\frac {2 d \left (b (c+d x) \text {Li}_2\left (-e^{a+b x}\right )-d \text {Li}_3\left (-e^{a+b x}\right )\right )}{b^2}+\frac {2 d \left (b (c+d x) \text {Li}_2\left (e^{a+b x}\right )-d \text {Li}_3\left (e^{a+b x}\right )\right )}{b^2}+(c+d x)^2 \log \left (1-e^{a+b x}\right )-(c+d x)^2 \log \left (e^{a+b x}+1\right )}{b} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.49, size = 242, normalized size = 2.44 \[ -\frac {2 \, d^{2} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, d^{2} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b d^{2} x + b c d\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (b d^{2} x + b c d\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{2} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 306, normalized size = 3.09 \[ -\frac {2 d^{2} a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 c^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}-\frac {2 d^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {2 d^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 c d \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 c d \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d^{2} \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 d^{2} \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 c d a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 c d \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}-\frac {2 c d \ln \left (1+{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {2 c d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {2 c d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 195, normalized size = 1.97 \[ -c^{2} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b}\right )} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{2}}{b^{3}} + \frac {{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{2} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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