Optimal. Leaf size=40 \[ \text {Int}\left (\frac {\coth (a+b x)}{x},x\right )+\frac {1}{2} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \]
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Rubi [A] time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx &=\int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx\\ &=\int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx\\ &=\frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx\\ &=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\int \frac {\coth (a+b x)}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 11.41, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \int \frac {1}{x e^{\left (b x + a\right )} + x}\,{d x} + \int \frac {1}{x e^{\left (b x + a\right )} - x}\,{d x} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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