Optimal. Leaf size=63 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)} \]
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Rubi [A] time = 0.05, 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 = {2212, 2211, 2204} \[ \frac {\sqrt {\pi } f^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rule 2212
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 {\operatorname {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 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 58, normalized size = 0.92 \[ -\frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} x \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) + 2 \, b f^{\frac {a x^{2} + b}{x^{2}}} \log \relax (f)}{4 \, b^{2} x \log \relax (f)^{2}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 58, normalized size = 0.92 \[ \frac {\sqrt {\pi }\, f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right )}{4 \sqrt {-b \ln \relax (f )}\, b \ln \relax (f )}-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 b x \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 28, normalized size = 0.44 \[ \frac {f^{a} \Gamma \left (\frac {3}{2}, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, x^{3} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {3}{2}}} \]
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 \[ \frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (f)}{x\,\sqrt {b\,\ln \relax (f)}}\right )}{4\,b\,\ln \relax (f)\,\sqrt {b\,\ln \relax (f)}}-\frac {f^a\,f^{\frac {b}{x^2}}}{2\,b\,x\,\ln \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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