Optimal. Leaf size=49 \[ b \cosh (a) \text {Chi}(b x)-\frac {\sinh (a+b x)}{x}+b \sinh (a) \text {Shi}(b x)-\text {Int}\left (\frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2},x\right ) \]
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Rubi [A]
time = 0.12, 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 {\sinh (a+b x) \tanh ^2(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 {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx &=\int \frac {\sinh (a+b x)}{x^2} \, dx-\int \frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2} \, dx\\ &=-\frac {\sinh (a+b x)}{x}+b \int \frac {\cosh (a+b x)}{x} \, dx-\int \frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2} \, dx\\ &=-\frac {\sinh (a+b x)}{x}+(b \cosh (a)) \int \frac {\cosh (b x)}{x} \, dx+(b \sinh (a)) \int \frac {\sinh (b x)}{x} \, dx-\int \frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2} \, dx\\ &=b \cosh (a) \text {Chi}(b x)-\frac {\sinh (a+b x)}{x}+b \sinh (a) \text {Shi}(b x)-\int \frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 6.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.88, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {sech}\left (b x +a \right )^{2} \left (\sinh ^{3}\left (b x +a \right )\right )}{x^{2}}\, dx\]
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 {\sinh ^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\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 {sinh}\left (a+b\,x\right )}^3}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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