Optimal. Leaf size=128 \[ \frac {3 \text {Li}_3\left (-\frac {a f^{2 x}}{b}\right )}{8 a b \log ^4(f)}-\frac {3 x \text {Li}_2\left (-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}-\frac {x^3}{2 a \log (f) \left (a f^{2 x}+b\right )}-\frac {3 x^2 \log \left (\frac {a f^{2 x}}{b}+1\right )}{4 a b \log ^2(f)}+\frac {x^3}{2 a b \log (f)} \]
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Rubi [A] time = 0.23, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2283, 2191, 2184, 2190, 2531, 2282, 6589} \[ -\frac {3 x \text {PolyLog}\left (2,-\frac {a f^{2 x}}{b}\right )}{4 a b \log ^3(f)}+\frac {3 \text {PolyLog}\left (3,-\frac {a f^{2 x}}{b}\right )}{8 a b \log ^4(f)}-\frac {3 x^2 \log \left (\frac {a f^{2 x}}{b}+1\right )}{4 a b \log ^2(f)}-\frac {x^3}{2 a \log (f) \left (a f^{2 x}+b\right )}+\frac {x^3}{2 a b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2191
Rule 2282
Rule 2283
Rule 2531
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b f^{-x}+a f^x\right )^2} \, dx &=\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 \operatorname {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*}
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Mathematica [A] time = 0.09, size = 124, normalized size = 0.97 \[ \frac {3 \left (\frac {\text {Li}_3\left (-\frac {a f^{2 x}}{b}\right )}{4 b \log ^3(f)}-\frac {x \text {Li}_2\left (-\frac {a f^{2 x}}{b}\right )}{2 b \log ^2(f)}-\frac {x^2 \log \left (\frac {a f^{2 x}}{b}+1\right )}{2 b \log (f)}+\frac {x^3}{3 b}\right )}{2 a \log (f)}-\frac {x^3}{2 a \log (f) \left (a f^{2 x}+b\right )} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.43, size = 241, normalized size = 1.88 \[ \frac {2 \, a f^{2 \, x} x^{3} \log \relax (f)^{3} - 6 \, {\left (a f^{2 \, x} x \log \relax (f) + b x \log \relax (f)\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {a}{b}}\right ) - 6 \, {\left (a f^{2 \, x} x \log \relax (f) + b x \log \relax (f)\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {a}{b}}\right ) - 3 \, {\left (a f^{2 \, x} x^{2} \log \relax (f)^{2} + b x^{2} \log \relax (f)^{2}\right )} \log \left (f^{x} \sqrt {-\frac {a}{b}} + 1\right ) - 3 \, {\left (a f^{2 \, x} x^{2} \log \relax (f)^{2} + b x^{2} \log \relax (f)^{2}\right )} \log \left (-f^{x} \sqrt {-\frac {a}{b}} + 1\right ) + 6 \, {\left (a f^{2 \, x} + b\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {a}{b}}\right ) + 6 \, {\left (a f^{2 \, x} + b\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {a}{b}}\right )}{4 \, {\left (a^{2} b f^{2 \, x} \log \relax (f)^{4} + a b^{2} \log \relax (f)^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (a f^{x} + \frac {b}{f^{x}}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 119, normalized size = 0.93 \[ -\frac {x^{3}}{2 \left (a \,f^{2 x}+b \right ) a \ln \relax (f )}+\frac {x^{3}}{2 a b \ln \relax (f )}-\frac {3 x^{2} \ln \left (\frac {a \,f^{2 x}}{b}+1\right )}{4 a b \ln \relax (f )^{2}}-\frac {3 x \polylog \left (2, -\frac {a \,f^{2 x}}{b}\right )}{4 a b \ln \relax (f )^{3}}+\frac {3 \polylog \left (3, -\frac {a \,f^{2 x}}{b}\right )}{8 a b \ln \relax (f )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 107, normalized size = 0.84 \[ -\frac {x^{3}}{2 \, {\left (a^{2} f^{2 \, x} \log \relax (f) + a b \log \relax (f)\right )}} + \frac {x^{3}}{2 \, a b \log \relax (f)} - \frac {3 \, {\left (2 \, x^{2} \log \left (\frac {a f^{2 \, x}}{b} + 1\right ) \log \relax (f)^{2} + 2 \, x {\rm Li}_2\left (-\frac {a f^{2 \, x}}{b}\right ) \log \relax (f) - {\rm Li}_{3}(-\frac {a f^{2 \, x}}{b})\right )}}{8 \, a b \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{{\left (\frac {b}{f^x}+a\,f^x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {x^{3}}{2 a^{2} f^{2 x} \log {\relax (f )} + 2 a b \log {\relax (f )}} + \frac {3 \int \frac {x^{2}}{a f^{2 x} + b}\, dx}{2 a \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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