Integrand size = 13, antiderivative size = 86 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=-\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac {5}{2}}(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^3 \log (f)} \]
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Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2243, 2242, 2235} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=-\frac {3 \sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac {5}{2}}(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^3 \log (f)} \]
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Rule 2235
Rule 2242
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+\frac {b}{x^2}}}{2 b x^3 \log (f)}-\frac {3 \int \frac {f^{a+\frac {b}{x^2}}}{x^4} \, dx}{2 b \log (f)} \\ & = \frac {3 f^{a+\frac {b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^3 \log (f)}+\frac {3 \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx}{4 b^2 \log ^2(f)} \\ & = \frac {3 f^{a+\frac {b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^3 \log (f)}-\frac {3 \text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )}{4 b^2 \log ^2(f)} \\ & = -\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac {5}{2}}(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^3 \log (f)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=-\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac {5}{2}}(f)}+\frac {f^{a+\frac {b}{x^2}} \left (3 x^2-2 b \log (f)\right )}{4 b^2 x^3 \log ^2(f)} \]
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Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
meijerg | \(\frac {f^{a} \sqrt {-b}\, \left (-\frac {\left (-b \right )^{\frac {5}{2}} \sqrt {\ln \left (f \right )}\, \left (-\frac {10 b \ln \left (f \right )}{x^{2}}+15\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{10 x \,b^{2}}+\frac {3 \left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{4 b^{\frac {5}{2}}}\right )}{2 \ln \left (f \right )^{\frac {5}{2}} b^{3}}\) | \(79\) |
risch | \(-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 x^{3} b \ln \left (f \right )}+\frac {3 f^{a} f^{\frac {b}{x^{2}}}}{4 \ln \left (f \right )^{2} b^{2} x}-\frac {3 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{8 \ln \left (f \right )^{2} b^{2} \sqrt {-b \ln \left (f \right )}}\) | \(80\) |
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Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=\frac {3 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} x^{3} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) + 2 \, {\left (3 \, b x^{2} \log \left (f\right ) - 2 \, b^{2} \log \left (f\right )^{2}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{8 \, b^{3} x^{3} \log \left (f\right )^{3}} \]
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Timed out. \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.33 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=\frac {f^{a} \Gamma \left (\frac {5}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{5} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {5}{2}}} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{6}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx=-\frac {f^a\,\left (3\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )-\frac {6\,f^{\frac {b}{x^2}}\,\sqrt {b\,\ln \left (f\right )}}{x}\right )}{8\,b^2\,{\ln \left (f\right )}^2\,\sqrt {b\,\ln \left (f\right )}}-\frac {f^a\,f^{\frac {b}{x^2}}}{2\,b\,x^3\,\ln \left (f\right )} \]
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