Optimal. Leaf size=57 \[ \frac {1}{2} b \sinh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {1}{2} b \cosh (2 a) \text {Shi}\left (2 b x^2\right )-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{4 x^2} \]
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Rubi [A] time = 0.13, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5341, 5321, 3297, 3303, 3298, 3301} \[ \frac {1}{2} b \sinh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {1}{2} b \cosh (2 a) \text {Shi}\left (2 b x^2\right )-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5321
Rule 5341
Rubi steps
\begin {align*} \int \frac {\cosh ^2\left (a+b x^2\right )}{x^3} \, dx &=\int \left (\frac {1}{2 x^3}+\frac {\cosh \left (2 a+2 b x^2\right )}{2 x^3}\right ) \, dx\\ &=-\frac {1}{4 x^2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^2\right )}{x^3} \, dx\\ &=-\frac {1}{4 x^2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\cosh (2 a+2 b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\sinh (2 a+2 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} (b \cosh (2 a)) \operatorname {Subst}\left (\int \frac {\sinh (2 b x)}{x} \, dx,x,x^2\right )+\frac {1}{2} (b \sinh (2 a)) \operatorname {Subst}\left (\int \frac {\cosh (2 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} b \text {Chi}\left (2 b x^2\right ) \sinh (2 a)+\frac {1}{2} b \cosh (2 a) \text {Shi}\left (2 b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 46, normalized size = 0.81 \[ \frac {1}{2} \left (b \sinh (2 a) \text {Chi}\left (2 b x^2\right )+b \cosh (2 a) \text {Shi}\left (2 b x^2\right )-\frac {\cosh ^2\left (a+b x^2\right )}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 90, normalized size = 1.58 \[ -\frac {\cosh \left (b x^{2} + a\right )^{2} - {\left (b x^{2} {\rm Ei}\left (2 \, b x^{2}\right ) - b x^{2} {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \cosh \left (2 \, a\right ) + \sinh \left (b x^{2} + a\right )^{2} - {\left (b x^{2} {\rm Ei}\left (2 \, b x^{2}\right ) + b x^{2} {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \sinh \left (2 \, a\right ) + 1}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 126, normalized size = 2.21 \[ \frac {2 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} - 2 \, a b^{2} {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} - 2 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + 2 \, a b^{2} {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} - b^{2} e^{\left (2 \, b x^{2} + 2 \, a\right )} - b^{2} e^{\left (-2 \, b x^{2} - 2 \, a\right )} - 2 \, b^{2}}{8 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 69, normalized size = 1.21 \[ -\frac {1}{4 x^{2}}-\frac {{\mathrm e}^{-2 a} {\mathrm e}^{-2 b \,x^{2}}}{8 x^{2}}+\frac {{\mathrm e}^{-2 a} b \Ei \left (1, 2 b \,x^{2}\right )}{4}-\frac {{\mathrm e}^{2 a} {\mathrm e}^{2 b \,x^{2}}}{8 x^{2}}-\frac {{\mathrm e}^{2 a} b \Ei \left (1, -2 b \,x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 36, normalized size = 0.63 \[ -\frac {1}{4} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x^{2}\right ) + \frac {1}{4} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x^{2}\right ) - \frac {1}{4 \, x^{2}} \]
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 \[ \int \frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\cosh ^{2}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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