Optimal. Leaf size=58 \[ \frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)-\frac {1}{4} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f) \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241}
\begin {gather*} -\frac {1}{4} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^2}\right )+\frac {1}{4} b x^2 \log (f) f^{a+\frac {b}{x^2}}+\frac {1}{4} x^4 f^{a+\frac {b}{x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2245
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^2}} x^3 \, dx &=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{2} (b \log (f)) \int f^{a+\frac {b}{x^2}} x \, dx\\ &=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx\\ &=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)-\frac {1}{4} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 44, normalized size = 0.76 \begin {gather*} \frac {1}{4} f^a \left (-b^2 \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f)+f^{\frac {b}{x^2}} x^2 \left (x^2+b \log (f)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 57, normalized size = 0.98
method | result | size |
risch | \(\frac {f^{a} x^{4} f^{\frac {b}{x^{2}}}}{4}+\frac {f^{a} \ln \left (f \right ) b \,x^{2} f^{\frac {b}{x^{2}}}}{4}+\frac {f^{a} \ln \left (f \right )^{2} b^{2} \expIntegral \left (1, -\frac {b \ln \left (f \right )}{x^{2}}\right )}{4}\) | \(57\) |
meijerg | \(-\frac {f^{a} b^{2} \ln \left (f \right )^{2} \left (\frac {x^{4} \left (\frac {9 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {12 b \ln \left (f \right )}{x^{2}}+6\right )}{12 b^{2} \ln \left (f \right )^{2}}-\frac {x^{4} \left (3+\frac {3 b \ln \left (f \right )}{x^{2}}\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{6 b^{2} \ln \left (f \right )^{2}}-\frac {\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{2}-\frac {\expIntegral \left (1, -\frac {b \ln \left (f \right )}{x^{2}}\right )}{2}-\frac {3}{4}-\ln \left (x \right )+\frac {\ln \left (-b \right )}{2}+\frac {\ln \left (\ln \left (f \right )\right )}{2}-\frac {x^{4}}{2 b^{2} \ln \left (f \right )^{2}}-\frac {x^{2}}{b \ln \left (f \right )}\right )}{2}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 22, normalized size = 0.38 \begin {gather*} \frac {1}{2} \, b^{2} f^{a} \Gamma \left (-2, -\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 47, normalized size = 0.81 \begin {gather*} -\frac {1}{4} \, b^{2} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{2} + \frac {1}{4} \, {\left (x^{4} + b x^{2} \log \left (f\right )\right )} f^{\frac {a x^{2} + b}{x^{2}}} \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 f^{a + \frac {b}{x^{2}}} x^{3}\, 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 = 3.65, size = 57, normalized size = 0.98 \begin {gather*} \frac {b^2\,f^a\,{\ln \left (f\right )}^2\,\left (f^{\frac {b}{x^2}}\,\left (\frac {x^2}{2\,b\,\ln \left (f\right )}+\frac {x^4}{2\,b^2\,{\ln \left (f\right )}^2}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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