Optimal. Leaf size=316 \[ -\frac{i x \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i x \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac{x^2 f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac{x^2 f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2}+\frac{x f^x}{4 a b \log ^2(f) \left (a f^{2 x}+b\right )} \]
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Rubi [A] time = 1.16332, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 14, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.737, Rules used = {2283, 2254, 2249, 199, 205, 2245, 14, 2282, 4848, 2391, 12, 5143, 2531, 6589} \[ -\frac{i x \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i x \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac{x^2 f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac{x^2 f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2}+\frac{x f^x}{4 a b \log ^2(f) \left (a f^{2 x}+b\right )} \]
Antiderivative was successfully verified.
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Rule 2283
Rule 2254
Rule 2249
Rule 199
Rule 205
Rule 2245
Rule 14
Rule 2282
Rule 4848
Rule 2391
Rule 12
Rule 5143
Rule 2531
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2}{\left (b f^{-x}+a f^x\right )^3} \, dx &=\int \frac{f^{3 x} x^2}{\left (b+a f^{2 x}\right )^3} \, dx\\ &=\int \left (-\frac{b f^x x^2}{a \left (b+a f^{2 x}\right )^3}+\frac{f^x x^2}{a \left (b+a f^{2 x}\right )^2}\right ) \, dx\\ &=\frac{\int \frac{f^x x^2}{\left (b+a f^{2 x}\right )^2} \, dx}{a}-\frac{b \int \frac{f^x x^2}{\left (b+a f^{2 x}\right )^3} \, dx}{a}\\ &=-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{2 \int x \left (\frac{f^x}{2 b \left (b+a f^{2 x}\right ) \log (f)}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2} \log (f)}\right ) \, dx}{a}+\frac{(2 b) \int x \left (\frac{f^x}{4 b \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{3 f^x}{8 b^2 \left (b+a f^{2 x}\right ) \log (f)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2} \log (f)}\right ) \, dx}{a}\\ &=-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{2 \int \left (\frac{f^x x}{2 b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2} \log (f)}\right ) \, dx}{a}+\frac{(2 b) \int \left (\frac{f^x x}{4 b \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{3 f^x x}{8 b^2 \left (b+a f^{2 x}\right ) \log (f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2} \log (f)}\right ) \, dx}{a}\\ &=-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac{\int \frac{f^x x}{\left (b+a f^{2 x}\right )^2} \, dx}{2 a \log (f)}+\frac{3 \int x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{4 a^{3/2} b^{3/2} \log (f)}-\frac{\int x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{a^{3/2} b^{3/2} \log (f)}+\frac{3 \int \frac{f^x x}{b+a f^{2 x}} \, dx}{4 a b \log (f)}-\frac{\int \frac{f^x x}{b+a f^{2 x}} \, dx}{a b \log (f)}\\ &=\frac{f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{\int \left (\frac{f^x}{2 b \left (b+a f^{2 x}\right ) \log (f)}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2} \log (f)}\right ) \, dx}{2 a \log (f)}+\frac{(3 i) \int x \log \left (1-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log (f)}-\frac{(3 i) \int x \log \left (1+\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log (f)}-\frac{i \int x \log \left (1-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log (f)}+\frac{i \int x \log \left (1+\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log (f)}-\frac{3 \int \frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \, dx}{4 a b \log (f)}+\frac{\int \frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \, dx}{a b \log (f)}\\ &=\frac{f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{i x \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i x \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}-\frac{(3 i) \int \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{(3 i) \int \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i \int \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log ^2(f)}-\frac{i \int \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log ^2(f)}-\frac{\int \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{4 a^{3/2} b^{3/2} \log ^2(f)}-\frac{3 \int \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{4 a^{3/2} b^{3/2} \log ^2(f)}+\frac{\int \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{a^{3/2} b^{3/2} \log ^2(f)}-\frac{\int \frac{f^x}{b+a f^{2 x}} \, dx}{4 a b \log ^2(f)}\\ &=\frac{f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{i x \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i x \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}-\frac{\operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}-\frac{3 \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac{\operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{a^{3/2} b^{3/2} \log ^3(f)}-\frac{\operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,f^x\right )}{4 a b \log ^3(f)}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac{f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{i x \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i x \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i \text{Li}_3\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \text{Li}_3\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^3(f)}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2} \log ^3(f)}+\frac{f^x x}{4 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac{f^x x^2}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac{f^x x^2}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac{i x \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i x \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}+\frac{i \text{Li}_3\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}-\frac{i \text{Li}_3\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2} \log ^3(f)}\\ \end{align*}
Mathematica [A] time = 0.467264, size = 254, normalized size = 0.8 \[ \frac{\frac{3 i \left (2 \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-2 \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-2 x \log (f) \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+2 x \log (f) \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+x^2 \log ^2(f) \log \left (1-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-x^2 \log ^2(f) \log \left (1+\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )\right )}{b^{3/2}}-\frac{12 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{12 \sqrt{a} x^2 f^x \log ^2(f)}{\left (a f^{2 x}+b\right )^2}+\frac{6 \sqrt{a} x f^x \log (f) (x \log (f)+2)}{b \left (a f^{2 x}+b\right )}}{48 a^{3/2} \log ^3(f)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\frac{b}{{f}^{x}}}+a{f}^{x} \right ) ^{-3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.67007, size = 1474, normalized size = 4.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{f^{- x} \left (a x^{2} \log{\left (f \right )} + 2 a x\right ) + f^{- 3 x} \left (- b x^{2} \log{\left (f \right )} + 2 b x\right )}{8 a^{3} b \log{\left (f \right )}^{2} + 16 a^{2} b^{2} f^{- 2 x} \log{\left (f \right )}^{2} + 8 a b^{3} f^{- 4 x} \log{\left (f \right )}^{2}} + \frac{\int - \frac{2 f^{x}}{a f^{2 x} + b}\, dx + \int \frac{f^{x} x^{2} \log{\left (f \right )}^{2}}{a f^{2 x} + b}\, dx}{8 a b \log{\left (f \right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (a f^{x} + \frac{b}{f^{x}}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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