Optimal. Leaf size=33 \[ \frac {\text {Li}_3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6610} \[ \frac {\text {PolyLog}\left (3,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6610
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac {\text {Li}_3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 32, normalized size = 0.97 \[ \frac {\text {Li}_3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 33, normalized size = 1.00 \[ \frac {{\rm polylog}\left (3, e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (b c - a d\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_2\left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (2, e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\right )} {\rm Li}_2\left (e e^{\left (n \log \left (b x + a\right ) - n \log \left (d x + c\right )\right )}\right ) + {\left (n \log \left (b x + a\right )^{2} - 2 \, n \log \left (b x + a\right ) \log \left (d x + c\right ) + n \log \left (d x + c\right )^{2}\right )} \log \left (-{\left (b x + a\right )}^{n} e + {\left (d x + c\right )}^{n}\right ) - {\left (n \log \left (b x + a\right )^{2} - 2 \, n \log \left (b x + a\right ) \log \left (d x + c\right ) + n \log \left (d x + c\right )^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, {\left (b c - a d\right )}} + \int -\frac {{\left (e n^{2} \log \left (b x + a\right )^{2} - 2 \, e n^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) + e n^{2} \log \left (d x + c\right )^{2}\right )} {\left (b x + a\right )}^{n}}{2 \, {\left ({\left (b d e x^{2} + a c e + {\left (b c + a d\right )} e x\right )} {\left (b x + a\right )}^{n} - {\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} {\left (d x + c\right )}^{n}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {polylog}\left (2,e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________