Optimal. Leaf size=62 \[ -\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \]
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Rubi [A] time = 0.07, antiderivative size = 62, 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 = {2212, 2209} \[ \frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \]
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^7} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}-\frac {2 \int \frac {f^{a+\frac {b}{x^2}}}{x^5} \, dx}{b \log (f)}\\ &=\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}+\frac {2 \int \frac {f^{a+\frac {b}{x^2}}}{x^3} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 45, normalized size = 0.73 \[ -\frac {f^{a+\frac {b}{x^2}} \left (b^2 \log ^2(f)-2 b x^2 \log (f)+2 x^4\right )}{2 b^3 x^4 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 0.76 \[ -\frac {{\left (2 \, x^{4} - 2 \, b x^{2} \log \relax (f) + b^{2} \log \relax (f)^{2}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{3} x^{4} \log \relax (f)^{3}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 74, normalized size = 1.19 \[ \frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b \ln \relax (f )}+\frac {x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}}{x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.30, size = 22, normalized size = 0.35 \[ -\frac {f^{a} \Gamma \left (3, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 47, normalized size = 0.76 \[ -\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \relax (f)}-\frac {x^2}{b^2\,{\ln \relax (f)}^2}+\frac {x^4}{b^3\,{\ln \relax (f)}^3}\right )}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 44, normalized size = 0.71 \[ \frac {f^{a + \frac {b}{x^{2}}} \left (- b^{2} \log {\relax (f )}^{2} + 2 b x^{2} \log {\relax (f )} - 2 x^{4}\right )}{2 b^{3} x^{4} \log {\relax (f )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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