Optimal. Leaf size=62 \[ -\frac {1}{4} b^2 \sinh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{4} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right )+\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {1}{4} b^2 \sinh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{4} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5320
Rubi steps
\begin {align*} \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x^3} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} \left (b^2 \cosh (a)\right ) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,\frac {1}{x^2}\right )-\frac {1}{4} \left (b^2 \sinh (a)\right ) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \text {Chi}\left (\frac {b}{x^2}\right ) \sinh (a)+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.90 \[ \frac {1}{4} \left (-b^2 \sinh (a) \text {Chi}\left (\frac {b}{x^2}\right )-b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right )+b x^2 \cosh \left (a+\frac {b}{x^2}\right )+x^4 \sinh \left (a+\frac {b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 89, normalized size = 1.44 \[ \frac {1}{4} \, x^{4} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \frac {1}{4} \, b x^{2} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \frac {1}{8} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x^{2}}\right ) - b^{2} {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \cosh \relax (a) - \frac {1}{8} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x^{2}}\right ) + b^{2} {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 353, normalized size = 5.69 \[ \frac {a^{2} b^{3} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )} - a^{2} b^{3} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a} - \frac {2 \, {\left (a x^{2} + b\right )} a b^{3} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )}}{x^{2}} + \frac {2 \, {\left (a x^{2} + b\right )} a b^{3} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a}}{x^{2}} - a b^{3} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )} - a b^{3} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )} + b^{3} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )} - b^{3} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )} + \frac {{\left (a x^{2} + b\right )}^{2} b^{3} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )}}{x^{4}} - \frac {{\left (a x^{2} + b\right )}^{2} b^{3} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a}}{x^{4}} + \frac {{\left (a x^{2} + b\right )} b^{3} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )}}{x^{2}} + \frac {{\left (a x^{2} + b\right )} b^{3} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )}}{x^{2}}}{8 \, {\left (a^{2} - \frac {2 \, {\left (a x^{2} + b\right )} a}{x^{2}} + \frac {{\left (a x^{2} + b\right )}^{2}}{x^{4}}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 93, normalized size = 1.50 \[ -\frac {{\mathrm e}^{-a} x^{4} {\mathrm e}^{-\frac {b}{x^{2}}}}{8}+\frac {{\mathrm e}^{-a} b \,x^{2} {\mathrm e}^{-\frac {b}{x^{2}}}}{8}-\frac {{\mathrm e}^{-a} b^{2} \Ei \left (1, \frac {b}{x^{2}}\right )}{8}+\frac {{\mathrm e}^{a} x^{4} {\mathrm e}^{\frac {b}{x^{2}}}}{8}+\frac {{\mathrm e}^{a} b \,{\mathrm e}^{\frac {b}{x^{2}}} x^{2}}{8}+\frac {{\mathrm e}^{a} b^{2} \Ei \left (1, -\frac {b}{x^{2}}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 44, normalized size = 0.71 \[ \frac {1}{4} \, x^{4} \sinh \left (a + \frac {b}{x^{2}}\right ) + \frac {1}{8} \, {\left (b e^{\left (-a\right )} \Gamma \left (-1, \frac {b}{x^{2}}\right ) - b e^{a} \Gamma \left (-1, -\frac {b}{x^{2}}\right )\right )} b \]
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 \[ \int x^3\,\mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]
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 \[ \int x^{3} \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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