Optimal. Leaf size=57 \[ \frac {e^{-a} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {e^a \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5454, 5406,
2235, 2236} \begin {gather*} \frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^a \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 5406
Rule 5454
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx &=-\text {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \text {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \text {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\pi } \left (\text {Erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))-\text {Erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a))\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 44, normalized size = 0.77
method | result | size |
risch | \(\frac {\erf \left (\frac {\sqrt {b}}{x}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{4 \sqrt {b}}-\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{4 \sqrt {-b}}\) | \(44\) |
meijerg | \(\frac {\sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \erf \left (\frac {\sqrt {b}}{x}\right )}{2 b^{\frac {3}{2}}}+\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \erfi \left (\frac {\sqrt {b}}{x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}+\frac {i \sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (\frac {\sqrt {i b}\, \sqrt {2}\, \erf \left (\frac {\sqrt {b}}{x}\right )}{2 \sqrt {b}}+\frac {\sqrt {i b}\, \sqrt {2}\, \erfi \left (\frac {\sqrt {b}}{x}\right )}{2 \sqrt {b}}\right )}{4 b}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 62, normalized size = 1.09 \begin {gather*} -\frac {1}{2} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (\frac {3}{2}, \frac {b}{x^{2}}\right )}{x^{3} \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}} + \frac {e^{a} \Gamma \left (\frac {3}{2}, -\frac {b}{x^{2}}\right )}{x^{3} \left (-\frac {b}{x^{2}}\right )^{\frac {3}{2}}}\right )} - \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 52, normalized size = 0.91 \begin {gather*} \frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) + \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \, b} \end {gather*}
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 \begin {gather*} \int \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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