Integrand size = 19, antiderivative size = 128 \[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=\frac {x^3}{2 a b \log (f)}-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac {3 x^2 \log \left (1+\frac {a f^{2 x}}{b}\right )}{4 a b \log ^2(f)}-\frac {3 x \operatorname {PolyLog}\left (2,-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}+\frac {3 \operatorname {PolyLog}\left (3,-\frac {a f^{2 x}}{b}\right )}{8 a b \log ^4(f)} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2321, 2222, 2215, 2221, 2611, 2320, 6724} \[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=\frac {3 \operatorname {PolyLog}\left (3,-\frac {a f^{2 x}}{b}\right )}{8 a b \log ^4(f)}-\frac {3 x \operatorname {PolyLog}\left (2,-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}-\frac {x^3}{2 a \log (f) \left (a f^{2 x}+b\right )}-\frac {3 x^2 \log \left (\frac {a f^{2 x}}{b}+1\right )}{4 a b \log ^2(f)}+\frac {x^3}{2 a b \log (f)} \]
[In]
[Out]
Rule 2215
Rule 2221
Rule 2222
Rule 2320
Rule 2321
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \frac {f^{2 x} x^3}{\left (b+a f^{2 x}\right )^2} \, dx \\ & = -\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}+\frac {3 \int \frac {x^2}{b+a f^{2 x}} \, dx}{2 a \log (f)} \\ & = \frac {x^3}{2 a b \log (f)}-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac {3 \int \frac {f^{2 x} x^2}{b+a f^{2 x}} \, dx}{2 b \log (f)} \\ & = \frac {x^3}{2 a b \log (f)}-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac {3 x^2 \log \left (1+\frac {a f^{2 x}}{b}\right )}{4 a b \log ^2(f)}+\frac {3 \int x \log \left (1+\frac {a f^{2 x}}{b}\right ) \, dx}{2 a b \log ^2(f)} \\ & = \frac {x^3}{2 a b \log (f)}-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac {3 x^2 \log \left (1+\frac {a f^{2 x}}{b}\right )}{4 a b \log ^2(f)}-\frac {3 x \text {Li}_2\left (-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}+\frac {3 \int \text {Li}_2\left (-\frac {a f^{2 x}}{b}\right ) \, dx}{4 a b \log ^3(f)} \\ & = \frac {x^3}{2 a b \log (f)}-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac {3 x^2 \log \left (1+\frac {a f^{2 x}}{b}\right )}{4 a b \log ^2(f)}-\frac {3 x \text {Li}_2\left (-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b}\right )}{x} \, dx,x,f^{2 x}\right )}{8 a b \log ^4(f)} \\ & = \frac {x^3}{2 a b \log (f)}-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac {3 x^2 \log \left (1+\frac {a f^{2 x}}{b}\right )}{4 a b \log ^2(f)}-\frac {3 x \text {Li}_2\left (-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}+\frac {3 \text {Li}_3\left (-\frac {a f^{2 x}}{b}\right )}{8 a b \log ^4(f)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97 \[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=-\frac {x^3}{2 a \left (b+a f^{2 x}\right ) \log (f)}+\frac {3 \left (\frac {x^3}{3 b}-\frac {x^2 \log \left (1+\frac {a f^{2 x}}{b}\right )}{2 b \log (f)}-\frac {x \operatorname {PolyLog}\left (2,-\frac {a f^{2 x}}{b}\right )}{2 b \log ^2(f)}+\frac {\operatorname {PolyLog}\left (3,-\frac {a f^{2 x}}{b}\right )}{4 b \log ^3(f)}\right )}{2 a \log (f)} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {x^{3}}{2 a b \ln \left (f \right )}-\frac {x^{3}}{2 a \left (b +a \,f^{2 x}\right ) \ln \left (f \right )}-\frac {3 x^{2} \ln \left (1+\frac {a \,f^{2 x}}{b}\right )}{4 a b \ln \left (f \right )^{2}}-\frac {3 x \,\operatorname {Li}_{2}\left (-\frac {a \,f^{2 x}}{b}\right )}{4 a b \ln \left (f \right )^{3}}+\frac {3 \,\operatorname {Li}_{3}\left (-\frac {a \,f^{2 x}}{b}\right )}{8 a b \ln \left (f \right )^{4}}\) | \(119\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (117) = 234\).
Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=\frac {2 \, a f^{2 \, x} x^{3} \log \left (f\right )^{3} - 6 \, {\left (a f^{2 \, x} x \log \left (f\right ) + b x \log \left (f\right )\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {a}{b}}\right ) - 6 \, {\left (a f^{2 \, x} x \log \left (f\right ) + b x \log \left (f\right )\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {a}{b}}\right ) - 3 \, {\left (a f^{2 \, x} x^{2} \log \left (f\right )^{2} + b x^{2} \log \left (f\right )^{2}\right )} \log \left (f^{x} \sqrt {-\frac {a}{b}} + 1\right ) - 3 \, {\left (a f^{2 \, x} x^{2} \log \left (f\right )^{2} + b x^{2} \log \left (f\right )^{2}\right )} \log \left (-f^{x} \sqrt {-\frac {a}{b}} + 1\right ) + 6 \, {\left (a f^{2 \, x} + b\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {a}{b}}\right ) + 6 \, {\left (a f^{2 \, x} + b\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {a}{b}}\right )}{4 \, {\left (a^{2} b f^{2 \, x} \log \left (f\right )^{4} + a b^{2} \log \left (f\right )^{4}\right )}} \]
[In]
[Out]
\[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=\frac {x^{3}}{2 a b \log {\left (f \right )} + 2 b^{2} f^{- 2 x} \log {\left (f \right )}} - \frac {3 \int \frac {f^{2 x} x^{2}}{a f^{2 x} + b}\, dx}{2 b \log {\left (f \right )}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=-\frac {x^{3}}{2 \, {\left (a^{2} f^{2 \, x} \log \left (f\right ) + a b \log \left (f\right )\right )}} + \frac {x^{3}}{2 \, a b \log \left (f\right )} - \frac {3 \, {\left (2 \, x^{2} \log \left (\frac {a f^{2 \, x}}{b} + 1\right ) \log \left (f\right )^{2} + 2 \, x {\rm Li}_2\left (-\frac {a f^{2 \, x}}{b}\right ) \log \left (f\right ) - {\rm Li}_{3}(-\frac {a f^{2 \, x}}{b})\right )}}{8 \, a b \log \left (f\right )^{4}} \]
[In]
[Out]
\[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=\int { \frac {x^{3}}{{\left (a f^{x} + \frac {b}{f^{x}}\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx=\int \frac {x^3}{{\left (\frac {b}{f^x}+a\,f^x\right )}^2} \,d x \]
[In]
[Out]