Optimal. Leaf size=67 \[ -\frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+x \sinh \left (a+\frac {b}{x^2}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5302, 5326, 5299, 2204, 2205} \[ -\frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )+x \sinh \left (a+\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5299
Rule 5302
Rule 5326
Rubi steps
\begin {align*} \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx &=-\operatorname {Subst}\left (\int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=x \sinh \left (a+\frac {b}{x^2}\right )-(2 b) \operatorname {Subst}\left (\int \cosh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=x \sinh \left (a+\frac {b}{x^2}\right )-b \operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-b \operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+x \sinh \left (a+\frac {b}{x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 70, normalized size = 1.04 \[ -\frac {1}{2} \sqrt {\pi } \sqrt {b} \left ((\cosh (a)-\sinh (a)) \text {erf}\left (\frac {\sqrt {b}}{x}\right )+(\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {\sqrt {b}}{x}\right )\right )+x \sinh (a) \cosh \left (\frac {b}{x^2}\right )+x \cosh (a) \sinh \left (\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 228, normalized size = 3.40 \[ \frac {x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + \sqrt {\pi } {\left (\cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - \sqrt {\pi } {\left (\cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) + 2 \, x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - x}{2 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh \left (a + \frac {b}{x^{2}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 1.04 \[ -\frac {\erf \left (\frac {\sqrt {b}}{x}\right ) {\mathrm e}^{-a} \sqrt {b}\, \sqrt {\pi }}{2}-\frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}} x}{2}+\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}} x}{2}-\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{2 \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 71, normalized size = 1.06 \[ -\frac {1}{2} \, b {\left (\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {b}{x^{2}}}\right ) - 1\right )} e^{\left (-a\right )}}{x \sqrt {\frac {b}{x^{2}}}} + \frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {b}{x^{2}}}\right ) - 1\right )} e^{a}}{x \sqrt {-\frac {b}{x^{2}}}}\right )} + x \sinh \left (a + \frac {b}{x^{2}}\right ) \]
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 \[ \int \mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]
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 \[ \int \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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