Optimal. Leaf size=25 \[ \frac {1}{2} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} \sinh (a) \text {Shi}\left (b x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5319, 5317, 5316} \[ \frac {1}{2} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} \sinh (a) \text {Shi}\left (b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 5316
Rule 5317
Rule 5319
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx &=\cosh (a) \int \frac {\cosh \left (b x^2\right )}{x} \, dx+\sinh (a) \int \frac {\sinh \left (b x^2\right )}{x} \, dx\\ &=\frac {1}{2} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} \sinh (a) \text {Shi}\left (b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.92 \[ \frac {1}{2} \left (\cosh (a) \text {Chi}\left (b x^2\right )+\sinh (a) \text {Shi}\left (b x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 39, normalized size = 1.56 \[ \frac {1}{4} \, {\left ({\rm Ei}\left (b x^{2}\right ) + {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \relax (a) + \frac {1}{4} \, {\left ({\rm Ei}\left (b x^{2}\right ) - {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 24, normalized size = 0.96 \[ \frac {1}{4} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{4} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 1.08 \[ -\frac {{\mathrm e}^{-a} \Ei \left (1, b \,x^{2}\right )}{4}-\frac {{\mathrm e}^{a} \Ei \left (1, -b \,x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 24, normalized size = 0.96 \[ \frac {1}{4} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{4} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \frac {\mathrm {cosh}\relax (a)\,\mathrm {coshint}\left (b\,x^2\right )}{2}+\frac {\mathrm {sinh}\relax (a)\,\mathrm {sinhint}\left (b\,x^2\right )}{2} \]
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 \[ \int \frac {\cosh {\left (a + b x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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