Optimal. Leaf size=92 \[ \frac {d \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}-\frac {d \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {d \text {csch}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4185, 4182, 2279, 2391} \[ \frac {d \text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}-\frac {d \text {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {d \text {csch}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 4185
Rubi steps
\begin {align*} \int (c+d x) \text {csch}^3(a+b x) \, dx &=-\frac {d \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int (c+d x) \text {csch}(a+b x) \, dx\\ &=\frac {(c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {d \int \log \left (1-e^{a+b x}\right ) \, dx}{2 b}-\frac {d \int \log \left (1+e^{a+b x}\right ) \, dx}{2 b}\\ &=\frac {(c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {d \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}-\frac {d \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=\frac {(c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x) \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {d \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}-\frac {d \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}\\ \end {align*}
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Mathematica [C] time = 2.31, size = 313, normalized size = 3.40 \[ -\frac {d \left (-a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )-i \left (i \left (\text {Li}_2\left (-e^{i (i a+i b x)}\right )-\text {Li}_2\left (e^{i (i a+i b x)}\right )\right )+(i a+i b x) \left (\log \left (1-e^{i (i a+i b x)}\right )-\log \left (1+e^{i (i a+i b x)}\right )\right )\right )\right )}{2 b^2}+\frac {d \text {csch}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right )}{4 b^2}+\frac {d \text {sech}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right )}{4 b^2}-\frac {c \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {c \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {c \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {d x \text {csch}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}-\frac {d x \text {sech}^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 1026, normalized size = 11.15 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 197, normalized size = 2.14 \[ -\frac {{\mathrm e}^{b x +a} \left (b d x \,{\mathrm e}^{2 b x +2 a}+b c \,{\mathrm e}^{2 b x +2 a}+b d x +d \,{\mathrm e}^{2 b x +2 a}+c b -d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {c \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}+\frac {d \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {d \ln \left (1+{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {d \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {d \polylog \left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}} \]
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 \[ -d {\left (\frac {{\left (b x e^{\left (3 \, a\right )} + e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + 8 \, \int \frac {x}{16 \, {\left (e^{\left (b x + a\right )} + 1\right )}}\,{d x} + 8 \, \int \frac {x}{16 \, {\left (e^{\left (b x + a\right )} - 1\right )}}\,{d x}\right )} + \frac {1}{2} \, c {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, {\left (e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}}\right )} \]
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 \[ \int \frac {c+d\,x}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \left (c + d x\right ) \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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