Optimal. Leaf size=52 \[ b \cosh (2 a) \text {Chi}(2 b x)-\frac {\sinh (2 a+2 b x)}{2 x}+b \sinh (2 a) \text {Shi}(2 b x)+\text {Int}\left (\frac {\coth (a+b x)}{x^2},x\right ) \]
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Rubi [A]
time = 0.11, 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 = {}
\begin {gather*} \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx \end {gather*}
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^2} \, dx &=\int \frac {\coth (a+b x)}{x^2} \, dx+\int \frac {\cosh (a+b x) \sinh (a+b x)}{x^2} \, dx\\ &=\int \frac {\coth (a+b x)}{x^2} \, dx+\int \frac {\sinh (2 a+2 b x)}{2 x^2} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x^2} \, dx+\int \frac {\coth (a+b x)}{x^2} \, dx\\ &=-\frac {\sinh (2 a+2 b x)}{2 x}+b \int \frac {\cosh (2 a+2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x^2} \, dx\\ &=-\frac {\sinh (2 a+2 b x)}{2 x}+(b \cosh (2 a)) \int \frac {\cosh (2 b x)}{x} \, dx+(b \sinh (2 a)) \int \frac {\sinh (2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x^2} \, dx\\ &=b \cosh (2 a) \text {Chi}(2 b x)-\frac {\sinh (2 a+2 b x)}{2 x}+b \sinh (2 a) \text {Shi}(2 b x)+\int \frac {\coth (a+b x)}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 11.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 2.85, size = 0, normalized size = 0.00 \[\int \frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x^2\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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