Optimal. Leaf size=55 \[ \frac {3}{8} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{8} \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3}{8} \sinh (a) \text {Shi}\left (b x^2\right )+\frac {1}{8} \sinh (3 a) \text {Shi}\left (3 b x^2\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5341, 5319, 5317, 5316} \[ \frac {3}{8} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{8} \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3}{8} \sinh (a) \text {Shi}\left (b x^2\right )+\frac {1}{8} \sinh (3 a) \text {Shi}\left (3 b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 5316
Rule 5317
Rule 5319
Rule 5341
Rubi steps
\begin {align*} \int \frac {\cosh ^3\left (a+b x^2\right )}{x} \, dx &=\int \left (\frac {3 \cosh \left (a+b x^2\right )}{4 x}+\frac {\cosh \left (3 a+3 b x^2\right )}{4 x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\cosh \left (3 a+3 b x^2\right )}{x} \, dx+\frac {3}{4} \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx\\ &=\frac {1}{4} (3 \cosh (a)) \int \frac {\cosh \left (b x^2\right )}{x} \, dx+\frac {1}{4} \cosh (3 a) \int \frac {\cosh \left (3 b x^2\right )}{x} \, dx+\frac {1}{4} (3 \sinh (a)) \int \frac {\sinh \left (b x^2\right )}{x} \, dx+\frac {1}{4} \sinh (3 a) \int \frac {\sinh \left (3 b x^2\right )}{x} \, dx\\ &=\frac {3}{8} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{8} \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3}{8} \sinh (a) \text {Shi}\left (b x^2\right )+\frac {1}{8} \sinh (3 a) \text {Shi}\left (3 b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 0.89 \[ \frac {1}{8} \left (3 \cosh (a) \text {Chi}\left (b x^2\right )+\cosh (3 a) \text {Chi}\left (3 b x^2\right )+3 \sinh (a) \text {Shi}\left (b x^2\right )+\sinh (3 a) \text {Shi}\left (3 b x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 83, normalized size = 1.51 \[ \frac {1}{16} \, {\left ({\rm Ei}\left (3 \, b x^{2}\right ) + {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) + \frac {3}{16} \, {\left ({\rm Ei}\left (b x^{2}\right ) + {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \relax (a) + \frac {1}{16} \, {\left ({\rm Ei}\left (3 \, b x^{2}\right ) - {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) + \frac {3}{16} \, {\left ({\rm Ei}\left (b x^{2}\right ) - {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 50, normalized size = 0.91 \[ \frac {1}{16} \, {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} + \frac {3}{16} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{16} \, {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + \frac {3}{16} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 55, normalized size = 1.00 \[ -\frac {{\mathrm e}^{-3 a} \Ei \left (1, 3 b \,x^{2}\right )}{16}-\frac {3 \,{\mathrm e}^{-a} \Ei \left (1, b \,x^{2}\right )}{16}-\frac {3 \,{\mathrm e}^{a} \Ei \left (1, -b \,x^{2}\right )}{16}-\frac {{\mathrm e}^{3 a} \Ei \left (1, -3 b \,x^{2}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 50, normalized size = 0.91 \[ \frac {1}{16} \, {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} + \frac {3}{16} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{16} \, {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + \frac {3}{16} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^3}{x} \,d x \]
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 \[ \int \frac {\cosh ^{3}{\left (a + b x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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