Optimal. Leaf size=87 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac {f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac {f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2282, 288, 199, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}+\frac {f^x}{8 a b \log (f) \left (a f^{2 x}+b\right )}-\frac {f^x}{4 a \log (f) \left (a f^{2 x}+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 288
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{\left (b f^{-x}+a f^x\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^2\right )^3} \, dx,x,f^x\right )}{\log (f)}\\ &=-\frac {f^x}{4 a \left (b+a f^{2 x}\right )^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 (f)}\\ &=-\frac {f^x}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,f^x\right )}{8 a b \log (f)}\\ &=-\frac {f^x}{4 a \left (b+a f^{2 x}\right )^2 \log (f)}+\frac {f^x}{8 a b \left (b+a f^{2 x}\right ) \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2} \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.80 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )+\frac {\sqrt {a} \sqrt {b} f^x \left (a f^{2 x}-b\right )}{\left (a f^{2 x}+b\right )^2}}{8 a^{3/2} b^{3/2} \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 261, normalized size = 3.00 \[ \left [\frac {2 \, a^{2} b f^{3 \, x} - 2 \, a b^{2} f^{x} - {\left (\sqrt {-a b} a^{2} f^{4 \, x} + 2 \, \sqrt {-a b} a b f^{2 \, x} + \sqrt {-a b} b^{2}\right )} \log \left (\frac {a f^{2 \, x} - 2 \, \sqrt {-a b} f^{x} - b}{a f^{2 \, x} + b}\right )}{16 \, {\left (a^{4} b^{2} f^{4 \, x} \log \relax (f) + 2 \, a^{3} b^{3} f^{2 \, x} \log \relax (f) + a^{2} b^{4} \log \relax (f)\right )}}, \frac {a^{2} b f^{3 \, x} - a b^{2} f^{x} - {\left (\sqrt {a b} a^{2} f^{4 \, x} + 2 \, \sqrt {a b} a b f^{2 \, x} + \sqrt {a b} b^{2}\right )} \arctan \left (\frac {\sqrt {a b}}{a f^{x}}\right )}{8 \, {\left (a^{4} b^{2} f^{4 \, x} \log \relax (f) + 2 \, a^{3} b^{3} f^{2 \, x} \log \relax (f) + a^{2} b^{4} \log \relax (f)\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 66, normalized size = 0.76 \[ \frac {\arctan \left (\frac {a f^{x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b \log \relax (f)} + \frac {a f^{3 \, x} - b f^{x}}{8 \, {\left (a f^{2 \, x} + b\right )}^{2} a b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 0.90 \[ -\frac {f^{x}}{8 \left (a \,f^{2 x}+b \right )^{2} a \ln \relax (f )}+\frac {f^{3 x}}{8 \left (a \,f^{2 x}+b \right )^{2} b \ln \relax (f )}+\frac {\arctan \left (\frac {a \,f^{x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 90, normalized size = 1.03 \[ -\frac {\frac {b}{f^{3 \, x}} - \frac {a}{f^{x}}}{8 \, {\left (a^{3} b + \frac {a b^{3}}{f^{4 \, x}} + \frac {2 \, a^{2} b^{2}}{f^{2 \, x}}\right )} \log \relax (f)} - \frac {\arctan \left (\frac {b}{\sqrt {a b} f^{x}}\right )}{8 \, \sqrt {a b} a b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.64, size = 113, normalized size = 1.30 \[ \frac {f^x}{8\,\left (a\,b^2\,\ln \relax (f)+a^2\,b\,f^{2\,x}\,\ln \relax (f)\right )}-\frac {f^x}{4\,\left (a\,b^2\,\ln \relax (f)+a^3\,f^{4\,x}\,\ln \relax (f)+2\,a^2\,b\,f^{2\,x}\,\ln \relax (f)\right )}+\frac {\mathrm {atan}\left (\frac {f^x\,\sqrt {a^3\,b^3\,{\ln \relax (f)}^2}}{a\,b^2\,\ln \relax (f)}\right )}{8\,\sqrt {a^3\,b^3\,{\ln \relax (f)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 85, normalized size = 0.98 \[ \frac {a f^{3 x} - b f^{x}}{8 a^{3} b f^{4 x} \log {\relax (f )} + 16 a^{2} b^{2} f^{2 x} \log {\relax (f )} + 8 a b^{3} \log {\relax (f )}} + \frac {\operatorname {RootSum} {\left (256 z^{2} a^{3} b^{3} + 1, \left (i \mapsto i \log {\left (16 i a b^{2} + f^{x} \right )} \right )\right )}}{\log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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