Optimal. Leaf size=42 \[ -\frac {1}{2} b \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{2} b \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x^2}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5320, 3297, 3303, 3298, 3301} \[ -\frac {1}{2} b \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{2} b \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5320
Rubi steps
\begin {align*} \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{2} (b \cosh (a)) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,\frac {1}{x^2}\right )-\frac {1}{2} (b \sinh (a)) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} b \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{2} b \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 0.93 \[ \frac {1}{2} \left (-b \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )+x^2 \sinh \left (a+\frac {b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 63, normalized size = 1.50 \[ \frac {1}{2} \, x^{2} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \frac {1}{4} \, {\left (b {\rm Ei}\left (\frac {b}{x^{2}}\right ) + b {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \cosh \relax (a) - \frac {1}{4} \, {\left (b {\rm Ei}\left (\frac {b}{x^{2}}\right ) - b {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 193, normalized size = 4.60 \[ -\frac {a b^{2} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )} - \frac {{\left (a x^{2} + b\right )} b^{2} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )}}{x^{2}} - b^{2} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )}}{4 \, {\left (a - \frac {a x^{2} + b}{x^{2}}\right )} b} - \frac {a b^{2} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a} - \frac {{\left (a x^{2} + b\right )} b^{2} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a}}{x^{2}} + b^{2} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )}}{4 \, {\left (a - \frac {a x^{2} + b}{x^{2}}\right )} b} \]
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 \[ -\frac {{\mathrm e}^{-a} x^{2} {\mathrm e}^{-\frac {b}{x^{2}}}}{4}+\frac {{\mathrm e}^{-a} b \Ei \left (1, \frac {b}{x^{2}}\right )}{4}+\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}} x^{2}}{4}+\frac {{\mathrm e}^{a} b \Ei \left (1, -\frac {b}{x^{2}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 39, normalized size = 0.93 \[ \frac {1}{2} \, x^{2} \sinh \left (a + \frac {b}{x^{2}}\right ) - \frac {1}{4} \, {\left ({\rm Ei}\left (-\frac {b}{x^{2}}\right ) e^{\left (-a\right )} + {\rm Ei}\left (\frac {b}{x^{2}}\right ) e^{a}\right )} b \]
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 \[ \int x\,\mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]
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 \[ \int x \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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