Integrand size = 16, antiderivative size = 172 \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)} \]
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Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2281, 205, 211, 2277, 2320, 4940, 2438} \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
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Rule 205
Rule 211
Rule 2277
Rule 2281
Rule 2320
Rule 2438
Rule 4940
Rubi steps \begin{align*} \text {integral}& = \frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\int \left (\frac {f^x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {\int \frac {f^x}{a+b f^{2 x}} \, dx}{2 a \log (f)}-\frac {\int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log (f)} \\ & = \frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,f^x\right )}{2 a \log ^2(f)}-\frac {\text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.58 \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=-\frac {\left (1+\frac {b f^{2 x}}{a}\right ) \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \left (a+b f^{2 x}\right ) \log ^2(f)}+\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {\frac {\frac {i x^2}{2 \sqrt {a}}-\frac {i x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}}{2 \sqrt {b}}+\frac {-\frac {i x^2}{2 \sqrt {a}}+\frac {i x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}}{2 \sqrt {b}}}{2 a} \]
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Time = 0.12 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {f^{x} x}{2 a \left (a +b \,f^{2 x}\right ) \ln \left (f \right )}+\frac {x \ln \left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 a \ln \left (f \right ) \sqrt {-a b}}-\frac {x \ln \left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 a \ln \left (f \right ) \sqrt {-a b}}+\frac {\operatorname {dilog}\left (\frac {-b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 a \ln \left (f \right )^{2} \sqrt {-a b}}-\frac {\operatorname {dilog}\left (\frac {b \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{4 a \ln \left (f \right )^{2} \sqrt {-a b}}-\frac {\arctan \left (\frac {b \,f^{x}}{\sqrt {a b}}\right )}{2 a \ln \left (f \right )^{2} \sqrt {a b}}\) | \(195\) |
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (120) = 240\).
Time = 0.27 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.81 \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {2 \, b f^{x} x \log \left (f\right ) + {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt {-\frac {b}{a}}\right ) + {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt {-\frac {b}{a}}\right ) - {\left (b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \left (f\right ) + a x \sqrt {-\frac {b}{a}} \log \left (f\right )\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left (b f^{2 \, x} x \sqrt {-\frac {b}{a}} \log \left (f\right ) + a x \sqrt {-\frac {b}{a}} \log \left (f\right )\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right )}{4 \, {\left (a b^{2} f^{2 \, x} \log \left (f\right )^{2} + a^{2} b \log \left (f\right )^{2}\right )}} \]
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\[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {f^{x} x}{2 a^{2} \log {\left (f \right )} + 2 a b f^{2 x} \log {\left (f \right )}} + \frac {\int \left (- \frac {f^{x}}{a + b f^{2 x}}\right )\, dx + \int \frac {f^{x} x \log {\left (f \right )}}{a + b f^{2 x}}\, dx}{2 a \log {\left (f \right )}} \]
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\[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=\int { \frac {f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{2}} \,d x } \]
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\[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=\int { \frac {f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx=\int \frac {f^x\,x}{{\left (a+b\,f^{2\,x}\right )}^2} \,d x \]
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