Optimal. Leaf size=50 \[ -\frac {d \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {d \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4182, 2279, 2391} \[ -\frac {d \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {d \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rubi steps
\begin {align*} \int (c+d x) \text {csch}(a+b x) \, dx &=-\frac {2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d \int \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {d \int \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {d \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}\\ &=-\frac {2 (c+d x) \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {d \text {Li}_2\left (e^{a+b x}\right )}{b^2}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 174, normalized size = 3.48 \[ \frac {d \left (-a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )-i \left (i \left (\text {Li}_2\left (-e^{i (i a+i b x)}\right )-\text {Li}_2\left (e^{i (i a+i b x)}\right )\right )+(i a+i b x) \left (\log \left (1-e^{i (i a+i b x)}\right )-\log \left (1+e^{i (i a+i b x)}\right )\right )\right )\right )}{b^2}+\frac {c \log \left (\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}-\frac {c \log \left (\cosh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 119, normalized size = 2.38 \[ \frac {d {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - d {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b d x + b c\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b c - a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b d x + a d\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{2}} \]
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 \[ \int {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 60, normalized size = 1.20 \[ \frac {\frac {d \left (2 \dilog \left ({\mathrm e}^{-b x -a}\right )-\frac {\dilog \left ({\mathrm e}^{-2 b x -2 a}\right )}{2}\right )}{b}+\frac {2 d a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}-2 c \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]
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 \[ -c {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b}\right )} + 2 \, d {\left (\int \frac {x}{2 \, {\left (e^{\left (b x + a\right )} + 1\right )}}\,{d x} + \int \frac {x}{2 \, {\left (e^{\left (b x + a\right )} - 1\right )}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {c+d\,x}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int \left (c + d x\right ) \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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