Optimal. Leaf size=25 \[ -\frac{1}{2} \sinh (a) \text{Chi}\left (\frac{b}{x^2}\right )-\frac{1}{2} \cosh (a) \text{Shi}\left (\frac{b}{x^2}\right ) \]
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Rubi [A] time = 0.0329483, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5318, 5317, 5316} \[ -\frac{1}{2} \sinh (a) \text{Chi}\left (\frac{b}{x^2}\right )-\frac{1}{2} \cosh (a) \text{Shi}\left (\frac{b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 5318
Rule 5317
Rule 5316
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+\frac{b}{x^2}\right )}{x} \, dx &=\cosh (a) \int \frac{\sinh \left (\frac{b}{x^2}\right )}{x} \, dx+\sinh (a) \int \frac{\cosh \left (\frac{b}{x^2}\right )}{x} \, dx\\ &=-\frac{1}{2} \text{Chi}\left (\frac{b}{x^2}\right ) \sinh (a)-\frac{1}{2} \cosh (a) \text{Shi}\left (\frac{b}{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0139663, size = 25, normalized size = 1. \[ \frac{1}{2} \left (\sinh (a) \left (-\text{Chi}\left (\frac{b}{x^2}\right )\right )-\cosh (a) \text{Shi}\left (\frac{b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 27, normalized size = 1.1 \begin{align*} -{\frac{{{\rm e}^{-a}}}{4}{\it Ei} \left ( 1,{\frac{b}{{x}^{2}}} \right ) }+{\frac{{{\rm e}^{a}}}{4}{\it Ei} \left ( 1,-{\frac{b}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38342, size = 32, normalized size = 1.28 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (-\frac{b}{x^{2}}\right ) e^{\left (-a\right )} - \frac{1}{4} \,{\rm Ei}\left (\frac{b}{x^{2}}\right ) e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70199, size = 105, normalized size = 4.2 \begin{align*} -\frac{1}{4} \,{\left ({\rm Ei}\left (\frac{b}{x^{2}}\right ) -{\rm Ei}\left (-\frac{b}{x^{2}}\right )\right )} \cosh \left (a\right ) - \frac{1}{4} \,{\left ({\rm Ei}\left (\frac{b}{x^{2}}\right ) +{\rm Ei}\left (-\frac{b}{x^{2}}\right )\right )} \sinh \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + \frac{b}{x^{2}} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (a + \frac{b}{x^{2}}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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