Integrand size = 16, antiderivative size = 51 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2347, 2209} \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]
[In]
[Out]
Rule 2209
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int \frac {e^{-\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n x^2} \\ & = \frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \text {Ei}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.71
method | result | size |
risch | \(-\frac {c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{b n}} \operatorname {Ei}_{1}\left (2 \ln \left (x \right )+\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{b n}\right )}{b n \,x^{2}}\) | \(240\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{2}}\right )}{b n} \]
[In]
[Out]
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
[In]
[Out]