Optimal. Leaf size=154 \[ -\frac {d^2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}+\frac {d^2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {d^2 \tanh ^{-1}(\cosh (a+b x))}{b^3}+\frac {d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}-\frac {d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {d (c+d x) \text {csch}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(c+d x)^2 \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.17, antiderivative size = 154, normalized size of antiderivative = 1.00, 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 2282
Rule 2531
Rule 3770
Rule 4182
Rule 4186
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*}
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Mathematica [B] time = 10.27, size = 420, normalized size = 2.73 \[ \frac {\text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c d \sinh \left (\frac {b x}{2}\right )+d^2 x \sinh \left (\frac {b x}{2}\right )\right )}{2 b^2}+\frac {\text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c d \sinh \left (\frac {b x}{2}\right )+d^2 x \sinh \left (\frac {b x}{2}\right )\right )}{2 b^2}-\frac {d \text {csch}(a) (c+d x)}{b^2}+\frac {b^2 \left (-c^2\right ) \log \left (1-e^{a+b x}\right )+b^2 c^2 \log \left (e^{a+b x}+1\right )-2 b^2 c d x \log \left (1-e^{a+b x}\right )+2 b^2 c d x \log \left (e^{a+b x}+1\right )-b^2 d^2 x^2 \log \left (1-e^{a+b x}\right )+b^2 d^2 x^2 \log \left (e^{a+b x}+1\right )+2 b d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )-2 b d (c+d x) \text {Li}_2\left (e^{a+b x}\right )-2 d^2 \text {Li}_3\left (-e^{a+b x}\right )+2 d^2 \text {Li}_3\left (e^{a+b x}\right )+2 d^2 \log \left (1-e^{a+b x}\right )-2 d^2 \log \left (e^{a+b x}+1\right )}{2 b^3}+\frac {\left (-c^2-2 c d x-d^2 x^2\right ) \text {csch}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {\left (-c^2-2 c d x-d^2 x^2\right ) \text {sech}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.55, size = 2218, normalized size = 14.40 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{2} \operatorname {csch}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 444, normalized size = 2.88 \[ -\frac {{\mathrm e}^{b x +a} \left (b \,d^{2} x^{2} {\mathrm e}^{2 b x +2 a}+2 b c d x \,{\mathrm e}^{2 b x +2 a}+b \,c^{2} {\mathrm e}^{2 b x +2 a}+b \,d^{2} x^{2}+2 d^{2} x \,{\mathrm e}^{2 b x +2 a}+2 b c d x +2 c d \,{\mathrm e}^{2 b x +2 a}+b \,c^{2}-2 d^{2} x -2 c d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {d^{2} \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {d^{2} \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 d^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}+\frac {d^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}-\frac {d^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {c d \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {c d \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 c d a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {c d \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}+\frac {c d \ln \left (1+{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {c d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}-\frac {c d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {d^{2} a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {c^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {d^{2} \ln \left (1+{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 393, normalized size = 2.55 \[ \frac {1}{2} \, c^{2} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, {\left (e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}}\right )} + \frac {{\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b d^{2} x^{2} e^{\left (3 \, a\right )} + 2 \, c d e^{\left (3 \, a\right )} + 2 \, {\left (b c d + d^{2}\right )} x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b d^{2} x^{2} e^{a} - 2 \, c d e^{a} + 2 \, {\left (b c d - d^{2}\right )} x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{2}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{2}}{2 \, b^{3}} - \frac {d^{2} \log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {d^{2} \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \left (c + d x\right )^{2} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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