Optimal. Leaf size=196 \[ -\frac {i \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{16 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{16 a^{3/2} b^{3/2} \log ^2(f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac {f^x}{8 a b \log ^2(f) \left (a f^{2 x}+b\right )}+\frac {x f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac {x f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2} \]
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Rubi [A] time = 0.50, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {2283, 2254, 2249, 199, 205, 2245, 2282, 4848, 2391} \[ -\frac {i \text {PolyLog}\left (2,-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{16 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \text {PolyLog}\left (2,\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{16 a^{3/2} b^{3/2} \log ^2(f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac {f^x}{8 a b \log ^2(f) \left (a f^{2 x}+b\right )}+\frac {x f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac {x f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 2245
Rule 2249
Rule 2254
Rule 2282
Rule 2283
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \frac {x}{\left (b f^{-x}+a f^x\right )^3} \, dx &=\int \frac {f^{3 x} x}{\left (b+a f^{2 x}\right )^3} \, dx\\ &=\int \left (-\frac {b f^x x}{a \left (b+a f^{2 x}\right )^3}+\frac {f^x x}{a \left (b+a f^{2 x}\right )^2}\right ) \, dx\\ &=\frac {\int \frac {f^x x}{\left (b+a f^{2 x}\right )^2} \, dx}{a}-\frac {b \int \frac {f^x x}{\left (b+a f^{2 x}\right )^3} \, dx}{a}\\ &=-\frac {f^x x}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x \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}{a}+\frac {b \int \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}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x \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}{\left (b+a f^{2 x}\right )^2} \, dx}{4 a \log (f)}+\frac {3 \int \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{8 a^{3/2} b^{3/2} \log (f)}-\frac {\int \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right ) \, dx}{2 a^{3/2} b^{3/2} \log (f)}+\frac {3 \int \frac {f^x}{b+a f^{2 x}} \, dx}{8 a b \log (f)}-\frac {\int \frac {f^x}{b+a f^{2 x}} \, dx}{2 a b \log (f)}\\ &=-\frac {f^x x}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (b+a x^2\right )^2} \, dx,x,f^x\right )}{4 a \log ^2(f)}+\frac {3 \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}-\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} b^{3/2} \log ^2(f)}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,f^x\right )}{8 a b \log ^2(f)}-\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,f^x\right )}{2 a b \log ^2(f)}\\ &=\frac {f^x}{8 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log ^2(f)}-\frac {f^x x}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (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 )}{16 a^{3/2} b^{3/2} \log ^2(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 )}{16 a^{3/2} b^{3/2} \log ^2(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 )}{4 a^{3/2} b^{3/2} \log ^2(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 )}{4 a^{3/2} b^{3/2} \log ^2(f)}+\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,f^x\right )}{8 a b \log ^2(f)}\\ &=\frac {f^x}{8 a b \left (b+a f^{2 x}\right ) \log ^2(f)}-\frac {f^x x}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{16 a^{3/2} b^{3/2} \log ^2(f)}+\frac {i \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{16 a^{3/2} b^{3/2} \log ^2(f)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 209, normalized size = 1.07 \[ \frac {-\frac {i \text {Li}_2\left (-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{b^{3/2}}+\frac {i \text {Li}_2\left (\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{b^{3/2}}+\frac {i x \log (f) \log \left (1-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {i x \log (f) \log \left (1+\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{b^{3/2}}+\frac {2 \sqrt {a} f^x}{a b f^{2 x}+b^2}+\frac {2 \sqrt {a} x f^x \log (f)}{a b f^{2 x}+b^2}-\frac {4 \sqrt {a} x f^x \log (f)}{\left (a f^{2 x}+b\right )^2}}{16 a^{3/2} \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 352, normalized size = 1.80 \[ \frac {2 \, {\left (a^{2} x \log \relax (f) + a^{2}\right )} f^{3 \, x} - 2 \, {\left (a b x \log \relax (f) - a b\right )} f^{x} + {\left (a^{2} f^{4 \, x} \sqrt {-\frac {a}{b}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {a}{b}} + b^{2} \sqrt {-\frac {a}{b}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {a}{b}}\right ) - {\left (a^{2} f^{4 \, x} \sqrt {-\frac {a}{b}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {a}{b}} + b^{2} \sqrt {-\frac {a}{b}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {a}{b}}\right ) - {\left (a^{2} f^{4 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + b^{2} x \sqrt {-\frac {a}{b}} \log \relax (f)\right )} \log \left (f^{x} \sqrt {-\frac {a}{b}} + 1\right ) + {\left (a^{2} f^{4 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + 2 \, a b f^{2 \, x} x \sqrt {-\frac {a}{b}} \log \relax (f) + b^{2} x \sqrt {-\frac {a}{b}} \log \relax (f)\right )} \log \left (-f^{x} \sqrt {-\frac {a}{b}} + 1\right )}{16 \, {\left (a^{4} b f^{4 \, x} \log \relax (f)^{2} + 2 \, a^{3} b^{2} f^{2 \, x} \log \relax (f)^{2} + a^{2} b^{3} \log \relax (f)^{2}\right )}} \]
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 \[ \int \frac {x}{{\left (a f^{x} + \frac {b}{f^{x}}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 209, normalized size = 1.07 \[ \frac {x \ln \left (\frac {-a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a b \ln \relax (f )}-\frac {x \ln \left (\frac {a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a b \ln \relax (f )}+\frac {\left (a x \,f^{2 x} \ln \relax (f )-b x \ln \relax (f )+a \,f^{2 x}+b \right ) f^{x}}{8 \left (a \,f^{2 x}+b \right )^{2} a b \ln \relax (f )^{2}}+\frac {\dilog \left (\frac {-a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a b \ln \relax (f )^{2}}-\frac {\dilog \left (\frac {a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{16 \sqrt {-a b}\, a b \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a x \log \relax (f) + a\right )} f^{3 \, x} - {\left (b x \log \relax (f) - b\right )} f^{x}}{8 \, {\left (a^{3} b f^{4 \, x} \log \relax (f)^{2} + 2 \, a^{2} b^{2} f^{2 \, x} \log \relax (f)^{2} + a b^{3} \log \relax (f)^{2}\right )}} + \int \frac {f^{x} x}{8 \, {\left (a^{2} b f^{2 \, x} + a b^{2}\right )}}\,{d x} \]
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 \[ \int \frac {x}{{\left (\frac {b}{f^x}+a\,f^x\right )}^3} \,d x \]
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 \[ \frac {f^{3 x} \left (a x \log {\relax (f )} + a\right ) + f^{x} \left (- b x \log {\relax (f )} + b\right )}{8 a^{3} b f^{4 x} \log {\relax (f )}^{2} + 16 a^{2} b^{2} f^{2 x} \log {\relax (f )}^{2} + 8 a b^{3} \log {\relax (f )}^{2}} + \frac {\int \frac {f^{x} x}{a f^{2 x} + b}\, dx}{8 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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