Optimal. Leaf size=44 \[ \frac {f^{a+\frac {b}{x^2}}}{2 b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^2 \log (f)} \]
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Rubi [A] time = 0.05, antiderivative size = 44, 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 = {2212, 2209} \[ \frac {f^{a+\frac {b}{x^2}}}{2 b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^2 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^5} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x^2 \log (f)}-\frac {\int \frac {f^{a+\frac {b}{x^2}}}{x^3} \, dx}{b \log (f)}\\ &=\frac {f^{a+\frac {b}{x^2}}}{2 b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^2 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.73 \[ \frac {f^{a+\frac {b}{x^2}} \left (x^2-b \log (f)\right )}{2 b^2 x^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 34, normalized size = 0.77 \[ \frac {{\left (x^{2} - b \log \relax (f)\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{2} x^{2} \log \relax (f)^{2}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 1.18 \[ \frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b \ln \relax (f )}+\frac {x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b^{2} \ln \relax (f )^{2}}}{x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.29, size = 22, normalized size = 0.50 \[ \frac {f^{a} \Gamma \left (2, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, b^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 36, normalized size = 0.82 \[ -\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \relax (f)}-\frac {x^2}{2\,b^2\,{\ln \relax (f)}^2}\right )}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 29, normalized size = 0.66 \[ \frac {f^{a + \frac {b}{x^{2}}} \left (- b \log {\relax (f )} + x^{2}\right )}{2 b^{2} x^{2} \log {\relax (f )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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