Integrand size = 13, antiderivative size = 58 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=-\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{6} b^2 f^a \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right ) \log ^2(f) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241} \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\frac {1}{6} b^2 f^a \log ^2(f) \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right )-\frac {b \log (f) f^{a+b x^3}}{6 x^3}-\frac {f^{a+b x^3}}{6 x^6} \]
[In]
[Out]
Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+b x^3}}{6 x^6}+\frac {1}{2} (b \log (f)) \int \frac {f^{a+b x^3}}{x^4} \, dx \\ & = -\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^3}}{x} \, dx \\ & = -\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{6} b^2 f^a \text {Ei}\left (b x^3 \log (f)\right ) \log ^2(f) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\frac {f^a \left (b^2 x^6 \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right ) \log ^2(f)-f^{b x^3} \left (1+b x^3 \log (f)\right )\right )}{6 x^6} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs. \(2(52)=104\).
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.43
method | result | size |
meijerg | \(\frac {f^{a} b^{2} \ln \left (f \right )^{2} \left (-\frac {1}{2 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {1}{b \,x^{3} \ln \left (f \right )}-\frac {3}{4}+\frac {3 \ln \left (x \right )}{2}+\frac {\ln \left (-b \right )}{2}+\frac {\ln \left (\ln \left (f \right )\right )}{2}+\frac {9 b^{2} x^{6} \ln \left (f \right )^{2}+12 b \,x^{3} \ln \left (f \right )+6}{12 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {\left (3+3 b \,x^{3} \ln \left (f \right )\right ) {\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{6 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {\ln \left (-b \,x^{3} \ln \left (f \right )\right )}{2}-\frac {\operatorname {Ei}_{1}\left (-b \,x^{3} \ln \left (f \right )\right )}{2}\right )}{3}\) | \(141\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\frac {b^{2} f^{a} x^{6} {\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} - {\left (b x^{3} \log \left (f\right ) + 1\right )} f^{b x^{3} + a}}{6 \, x^{6}} \]
[In]
[Out]
\[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\int \frac {f^{a + b x^{3}}}{x^{7}}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.38 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=-\frac {1}{3} \, b^{2} f^{a} \Gamma \left (-2, -b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} \]
[In]
[Out]
\[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\int { \frac {f^{b x^{3} + a}}{x^{7}} \,d x } \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=-\frac {b^2\,f^a\,{\ln \left (f\right )}^2\,\left (f^{b\,x^3}\,\left (\frac {1}{2\,b\,x^3\,\ln \left (f\right )}+\frac {1}{2\,b^2\,x^6\,{\ln \left (f\right )}^2}\right )+\frac {\mathrm {expint}\left (-b\,x^3\,\ln \left (f\right )\right )}{2}\right )}{3} \]
[In]
[Out]