Optimal. Leaf size=37 \[ \frac {1}{4} \cosh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {1}{4} \sinh (2 a) \text {Shi}\left (2 b x^2\right )-\frac {\log (x)}{2} \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5340, 5319, 5317, 5316} \[ \frac {1}{4} \cosh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {1}{4} \sinh (2 a) \text {Shi}\left (2 b x^2\right )-\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 5316
Rule 5317
Rule 5319
Rule 5340
Rubi steps
\begin {align*} \int \frac {\sinh ^2\left (a+b x^2\right )}{x} \, dx &=\int \left (-\frac {1}{2 x}+\frac {\cosh \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx\\ &=-\frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^2\right )}{x} \, dx\\ &=-\frac {\log (x)}{2}+\frac {1}{2} \cosh (2 a) \int \frac {\cosh \left (2 b x^2\right )}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\sinh \left (2 b x^2\right )}{x} \, dx\\ &=\frac {1}{4} \cosh (2 a) \text {Chi}\left (2 b x^2\right )-\frac {\log (x)}{2}+\frac {1}{4} \sinh (2 a) \text {Shi}\left (2 b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.89 \[ \frac {1}{4} \left (\cosh (2 a) \text {Chi}\left (2 b x^2\right )+\sinh (2 a) \text {Shi}\left (2 b x^2\right )-2 \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 49, normalized size = 1.32 \[ \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x^{2}\right ) + {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x^{2}\right ) - {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \sinh \left (2 \, a\right ) - \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 35, normalized size = 0.95 \[ \frac {1}{8} \, {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} - \frac {1}{4} \, \log \left (b x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 34, normalized size = 0.92 \[ -\frac {\ln \relax (x )}{2}-\frac {{\mathrm e}^{-2 a} \Ei \left (1, 2 b \,x^{2}\right )}{8}-\frac {{\mathrm e}^{2 a} \Ei \left (1, -2 b \,x^{2}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 31, normalized size = 0.84 \[ \frac {1}{8} \, {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} - \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^2}{x} \,d x \]
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 \[ \int \frac {\sinh ^{2}{\left (a + b x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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