Optimal. Leaf size=268 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
[Out]
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Rubi [A] time = 0.3977, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
Antiderivative was successfully verified.
[In] Int[x^3/(b/f^x + a*f^x),x]
[Out]
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Rubi in Sympy [A] time = 102.562, size = 264, normalized size = 0.99 \[ - \frac{x^{3} \operatorname{atan}{\left (\frac{\sqrt{b} f^{- x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} \log{\left (f \right )}} - \frac{3 i x^{2} \operatorname{Li}_{2}\left (- \frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{3 i x^{2} \operatorname{Li}_{2}\left (\frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} - \frac{3 i x \operatorname{Li}_{3}\left (- \frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} + \frac{3 i x \operatorname{Li}_{3}\left (\frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} - \frac{3 i \operatorname{Li}_{4}\left (- \frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{4}} + \frac{3 i \operatorname{Li}_{4}\left (\frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b/(f**x)+a*f**x),x)
[Out]
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Mathematica [A] time = 0.0521668, size = 224, normalized size = 0.84 \[ \frac{i \left (-3 x^2 \log ^2(f) \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+3 x^2 \log ^2(f) \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-6 \text{PolyLog}\left (4,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+6 \text{PolyLog}\left (4,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+6 x \log (f) \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-6 x \log (f) \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+x^3 \log ^3(f) \log \left (1-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-x^3 \log ^3(f) \log \left (1+\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )\right )}{2 \sqrt{a} \sqrt{b} \log ^4(f)} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(b/f^x + a*f^x),x]
[Out]
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Maple [F] time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{3} \left ({\frac{b}{{f}^{x}}}+a{f}^{x} \right ) ^{-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b/(f^x)+a*f^x),x)
[Out]
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a*f^x + b/f^x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255991, size = 433, normalized size = 1.62 \[ \frac{x^{3} \sqrt{-\frac{a}{b}} \log \left (f\right )^{3} \log \left (\frac{a f^{x} + b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}}\right ) - x^{3} \sqrt{-\frac{a}{b}} \log \left (f\right )^{3} \log \left (-\frac{a f^{x} - b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}}\right ) + 3 \, x^{2} \sqrt{-\frac{a}{b}}{\rm Li}_2\left (-\frac{a f^{x} + b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}} + 1\right ) \log \left (f\right )^{2} - 3 \, x^{2} \sqrt{-\frac{a}{b}}{\rm Li}_2\left (\frac{a f^{x} - b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}} + 1\right ) \log \left (f\right )^{2} + 6 \, x \sqrt{-\frac{a}{b}} \log \left (f\right ){\rm Li}_{3}(\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}}) - 6 \, x \sqrt{-\frac{a}{b}} \log \left (f\right ){\rm Li}_{3}(-\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}}) - 6 \, \sqrt{-\frac{a}{b}}{\rm Li}_{4}(\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}}) + 6 \, \sqrt{-\frac{a}{b}}{\rm Li}_{4}(-\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}})}{2 \, a \log \left (f\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a*f^x + b/f^x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x^{3}}{a f^{2 x} + b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b/(f**x)+a*f**x),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{a f^{x} + \frac{b}{f^{x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a*f^x + b/f^x),x, algorithm="giac")
[Out]