Optimal. Leaf size=74 \[ \frac {d^2 \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \coth (a+b x)}{b}-\frac {(c+d x)^2}{b} \]
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Rubi [A] time = 0.14, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4184, 3716, 2190, 2279, 2391} \[ \frac {d^2 \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \coth (a+b x)}{b}-\frac {(c+d x)^2}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 4184
Rubi steps
\begin {align*} \int (c+d x)^2 \text {csch}^2(a+b x) \, dx &=-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {(2 d) \int (c+d x) \coth (a+b x) \, dx}{b}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}-\frac {(4 d) \int \frac {e^{2 (a+b x)} (c+d x)}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {\left (2 d^2\right ) \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{b^3}\\ &=-\frac {(c+d x)^2}{b}-\frac {(c+d x)^2 \coth (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {d^2 \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [C] time = 5.45, size = 198, normalized size = 2.68 \[ \frac {\text {csch}(a) \left (b^2 \sinh (b x) (c+d x)^2 \text {csch}(a+b x)+d^2 \left (-b^2 x^2 \cosh (a) e^{-\tanh ^{-1}(\tanh (a))} \sqrt {\text {sech}^2(a)}-\sinh (a) \text {Li}_2\left (e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )+i \pi b x \sinh (a)-i \pi \sinh (a) \log \left (e^{2 b x}+1\right )+2 b x \sinh (a) \log \left (1-e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )+2 \sinh (a) \tanh ^{-1}(\tanh (a)) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}(\tanh (a))+b x\right )\right )+b x\right )+i \pi \sinh (a) \log (\cosh (b x))\right )-2 b c d (b x \cosh (a)-\sinh (a) \log (\sinh (a+b x)))\right )}{b^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 2.29, size = 623, normalized size = 8.42 \[ -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \sinh \left (b x + a\right )^{2} - {\left (d^{2} \cosh \left (b x + a\right )^{2} + 2 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} - d^{2}\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\left (d^{2} \cosh \left (b x + a\right )^{2} + 2 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} - d^{2}\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b d^{2} x + b c d - {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b c d - a d^{2} - {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b c d - a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b d^{2} x + a d^{2} - {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b d^{2} x + a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - 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 \[ \int {\left (d x + c\right )}^{2} \operatorname {csch}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 240, normalized size = 3.24 \[ -\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {4 d c \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}+\frac {2 d c \ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 d^{2} x^{2}}{b}-\frac {4 d^{2} a x}{b^{2}}-\frac {2 d^{2} a^{2}}{b^{3}}+\frac {2 d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 d^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 d^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 d^{2} a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, d^{2} {\left (\frac {x^{2}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} + 2 \, \int \frac {x}{2 \, {\left (b e^{\left (b x + a\right )} + b\right )}}\,{d x} - 2 \, \int \frac {x}{2 \, {\left (b e^{\left (b x + a\right )} - b\right )}}\,{d x}\right )} - 2 \, c d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac {2 \, c^{2}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \left (c + d x\right )^{2} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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