Optimal. Leaf size=39 \[ -\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2242, 2235}
\begin {gather*} -\frac {\sqrt {\pi } f^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2242
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx &=-\text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.00 \begin {gather*} -\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 28, normalized size = 0.72
method | result | size |
meijerg | \(-\frac {f^{a} \erfi \left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right ) \sqrt {\pi }}{2 \sqrt {b}\, \sqrt {\ln \left (f \right )}}\) | \(28\) |
risch | \(-\frac {f^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{2 \sqrt {-b \ln \left (f \right )}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 34, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {\pi } f^{a} {\left (\operatorname {erf}\left (\sqrt {-\frac {b \log \left (f\right )}{x^{2}}}\right ) - 1\right )}}{2 \, x \sqrt {-\frac {b \log \left (f\right )}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 34, normalized size = 0.87 \begin {gather*} \frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right )}{2 \, b \log \left (f\right )} \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 {f^{a + \frac {b}{x^{2}}}}{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 [B]
time = 3.52, size = 28, normalized size = 0.72 \begin {gather*} -\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )}{2\,\sqrt {b\,\ln \left (f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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