Integrand size = 12, antiderivative size = 50 \[ \int (c+d x) \text {csch}(a+b x) \, dx=-\frac {2 (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4267, 2317, 2438} \[ \int (c+d x) \text {csch}(a+b x) \, dx=-\frac {2 (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2} \]
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Rule 2317
Rule 2438
Rule 4267
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x) \text {arctanh}\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) \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {d \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^2} \\ & = -\frac {2 (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(116\) vs. \(2(50)=100\).
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.32 \[ \int (c+d x) \text {csch}(a+b x) \, dx=-\frac {c \log \left (\cosh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}+\frac {c \log \left (\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}+2 d \left (\frac {x \log \left (1-e^{a+b x}\right )}{2 b}-\frac {x \log \left (1+e^{a+b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {\operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.78
method | result | size |
derivativedivides | \(\frac {\frac {d \left (\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )-\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )\right )}{b}+\frac {2 d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-2 c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(89\) |
default | \(\frac {\frac {d \left (\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )-\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )\right )}{b}+\frac {2 d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}-2 c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(89\) |
parts | \(\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d x}{b}+\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c}{b}+\frac {2 d \left (-\frac {\operatorname {dilog}\left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}-\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \ln \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}-\frac {\operatorname {dilog}\left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}\right )}{b^{2}}\) | \(90\) |
risch | \(-\frac {2 c \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}+\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {d \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {d \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {d \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}-\frac {d \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}-\frac {d \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(124\) |
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (45) = 90\).
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.38 \[ \int (c+d x) \text {csch}(a+b x) \, dx=\frac {d {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - d {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b d x + b c\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + {\left (b c - a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b d x + a d\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{2}} \]
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\[ \int (c+d x) \text {csch}(a+b x) \, dx=\int \left (c + d x\right ) \operatorname {csch}{\left (a + b x \right )}\, dx \]
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\[ \int (c+d x) \text {csch}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right ) \,d x } \]
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\[ \int (c+d x) \text {csch}(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {csch}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x) \text {csch}(a+b x) \, dx=\int \frac {c+d\,x}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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