Optimal. Leaf size=103 \[ -\frac {3 d^3 \text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {(c+d x)^3}{b} \]
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Rubi [A] time = 0.23, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4184, 3716, 2190, 2531, 2282, 6589} \[ \frac {3 d^2 (c+d x) \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}-\frac {3 d^3 \text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {(c+d x)^3}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 4184
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^3 \text {csch}^2(a+b x) \, dx &=-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {(3 d) \int (c+d x)^2 \coth (a+b x) \, dx}{b}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {(6 d) \int \frac {e^{2 (a+b x)} (c+d x)^2}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac {\left (3 d^3\right ) \int \text {Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac {\left (3 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac {3 d^3 \text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 1.97, size = 185, normalized size = 1.80 \[ \frac {-\frac {2 b^3 (c+d x)^3}{e^{2 a}-1}+b^3 \text {csch}(a) \sinh (b x) (c+d x)^3 \text {csch}(a+b x)+3 b^2 d (c+d x)^2 \log \left (1-e^{-a-b x}\right )+3 b^2 d (c+d x)^2 \log \left (e^{-a-b x}+1\right )-6 d^2 \left (b (c+d x) \text {Li}_2\left (-e^{-a-b x}\right )+d \text {Li}_3\left (-e^{-a-b x}\right )\right )-6 d^2 \left (b (c+d x) \text {Li}_2\left (e^{-a-b x}\right )+d \text {Li}_3\left (e^{-a-b x}\right )\right )}{b^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.61, size = 1159, normalized size = 11.25 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{3} \operatorname {csch}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 473, normalized size = 4.59 \[ -\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {6 d^{3} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {6 d^{2} c \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{2} c \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{3} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {6 d^{2} c \,x^{2}}{b}-\frac {6 d^{2} c \,a^{2}}{b^{3}}+\frac {6 d^{3} a^{2} x}{b^{3}}-\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {3 d^{3} \ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {4 d^{3} a^{3}}{b^{4}}-\frac {2 d^{3} x^{3}}{b}-\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 d^{2} c a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {12 d^{2} c a x}{b^{2}}+\frac {6 d^{2} c \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {12 d^{2} c a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.80, size = 320, normalized size = 3.11 \[ -3 \, c^{2} d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac {6 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d^{2}}{b^{3}} + \frac {6 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d^{2}}{b^{3}} + \frac {2 \, c^{3}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} - \frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2}\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{3}}{b^{4}} - \frac {2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2}\right )}}{b^{4}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^3}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \left (c + d x\right )^{3} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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