Optimal. Leaf size=42 \[ \frac {1}{2} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} b \sinh (a) \text {Shi}\left (b x^2\right )-\frac {\sinh \left (a+b x^2\right )}{2 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5320, 3297, 3303, 3298, 3301} \[ \frac {1}{2} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} b \sinh (a) \text {Shi}\left (b x^2\right )-\frac {\sinh \left (a+b x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5320
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b x^2\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sinh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {\sinh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} (b \cosh (a)) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{2} (b \sinh (a)) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} b \cosh (a) \text {Chi}\left (b x^2\right )-\frac {\sinh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} b \sinh (a) \text {Shi}\left (b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 38, normalized size = 0.90 \[ \frac {1}{2} \left (b \cosh (a) \text {Chi}\left (b x^2\right )+b \sinh (a) \text {Shi}\left (b x^2\right )-\frac {\sinh \left (a+b x^2\right )}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 71, normalized size = 1.69 \[ \frac {{\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) + b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \relax (a) + {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) - b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \relax (a) - 2 \, \sinh \left (b x^{2} + a\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 109, normalized size = 2.60 \[ \frac {{\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - a b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - a b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (b x^{2} + a\right )} + b^{2} e^{\left (-b x^{2} - a\right )}}{4 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 58, normalized size = 1.38 \[ \frac {{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{4 x^{2}}-\frac {{\mathrm e}^{-a} b \Ei \left (1, b \,x^{2}\right )}{4}-\frac {{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{4 x^{2}}-\frac {{\mathrm e}^{a} b \Ei \left (1, -b \,x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 39, normalized size = 0.93 \[ \frac {1}{4} \, {\left ({\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + {\rm Ei}\left (b x^{2}\right ) e^{a}\right )} b - \frac {\sinh \left (b x^{2} + a\right )}{2 \, x^{2}} \]
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 \[ \int \frac {\mathrm {sinh}\left (b\,x^2+a\right )}{x^3} \,d x \]
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 \[ \int \frac {\sinh {\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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