Optimal. Leaf size=91 \[ -\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
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Rubi [A] time = 0.21, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5340, 5320, 3297, 3303, 3298, 3301} \[ -\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5320
Rule 5340
Rubi steps
\begin {align*} \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx &=\int \left (-\frac {3 \sinh \left (a+b x^2\right )}{4 x^3}+\frac {\sinh \left (3 a+3 b x^2\right )}{4 x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh \left (3 a+3 b x^2\right )}{x^3} \, dx-\frac {3}{4} \int \frac {\sinh \left (a+b x^2\right )}{x^3} \, dx\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {\sinh (3 a+3 b x)}{x^2} \, dx,x,x^2\right )-\frac {3}{8} \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {\cosh (3 a+3 b x)}{x} \, dx,x,x^2\right )\\ &=\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {1}{8} (3 b \cosh (a)) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \cosh (3 a)) \operatorname {Subst}\left (\int \frac {\cosh (3 b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \sinh (a)) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \sinh (3 a)) \operatorname {Subst}\left (\int \frac {\sinh (3 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 90, normalized size = 0.99 \[ -\frac {3 b x^2 \cosh (a) \text {Chi}\left (b x^2\right )-3 b x^2 \cosh (3 a) \text {Chi}\left (3 b x^2\right )+3 b x^2 \sinh (a) \text {Shi}\left (b x^2\right )-3 b x^2 \sinh (3 a) \text {Shi}\left (3 b x^2\right )-3 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 160, normalized size = 1.76 \[ -\frac {2 \, \sinh \left (b x^{2} + a\right )^{3} - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) + b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) + 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) + b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \relax (a) + 6 \, {\left (\cosh \left (b x^{2} + a\right )^{2} - 1\right )} \sinh \left (b x^{2} + a\right ) - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) - b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) + 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) - b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \relax (a)}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 223, normalized size = 2.45 \[ \frac {3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, a b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} + 3 \, a b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (3 \, b x^{2} + 3 \, a\right )} + 3 \, b^{2} e^{\left (b x^{2} + a\right )} - 3 \, b^{2} e^{\left (-b x^{2} - a\right )} + b^{2} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{16 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 120, normalized size = 1.32 \[ \frac {{\mathrm e}^{-3 a} {\mathrm e}^{-3 b \,x^{2}}}{16 x^{2}}-\frac {3 \,{\mathrm e}^{-3 a} b \Ei \left (1, 3 b \,x^{2}\right )}{16}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{16 x^{2}}+\frac {3 \,{\mathrm e}^{-a} b \Ei \left (1, b \,x^{2}\right )}{16}-\frac {{\mathrm e}^{3 a} {\mathrm e}^{3 b \,x^{2}}}{16 x^{2}}-\frac {3 \,{\mathrm e}^{3 a} b \Ei \left (1, -3 b \,x^{2}\right )}{16}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{16 x^{2}}+\frac {3 \,{\mathrm e}^{a} b \Ei \left (1, -b \,x^{2}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 58, normalized size = 0.64 \[ \frac {3}{16} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{2}\right ) - \frac {3}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{2}\right ) - \frac {3}{16} \, b e^{a} \Gamma \left (-1, -b x^{2}\right ) + \frac {3}{16} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\sinh ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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