Optimal. Leaf size=63 \[ \frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)} \]
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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2243, 2242,
2235} \begin {gather*} \frac {\sqrt {\pi } f^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2242
Rule 2243
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^4} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)}-\frac {\int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx}{2 b \log (f)}\\ &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)}+\frac {\text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )}{2 b \log (f)}\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 63, normalized size = 1.00 \begin {gather*} \frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 58, normalized size = 0.92
method | result | size |
risch | \(-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 x b \ln \left (f \right )}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{4 \ln \left (f \right ) b \sqrt {-b \ln \left (f \right )}}\) | \(58\) |
meijerg | \(-\frac {f^{a} \sqrt {-b}\, \left (\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (f \right )}\, {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{x b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \erfi \left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{2 b^{\frac {3}{2}}}\right )}{2 \ln \left (f \right )^{\frac {3}{2}} b^{2}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 28, normalized size = 0.44 \begin {gather*} \frac {f^{a} \Gamma \left (\frac {3}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{3} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 58, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} x \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) + 2 \, b f^{\frac {a x^{2} + b}{x^{2}}} \log \left (f\right )}{4 \, b^{2} x \log \left (f\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 3.56, size = 58, normalized size = 0.92 \begin {gather*} \frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )}{4\,b\,\ln \left (f\right )\,\sqrt {b\,\ln \left (f\right )}}-\frac {f^a\,f^{\frac {b}{x^2}}}{2\,b\,x\,\ln \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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