Optimal. Leaf size=74 \[ \frac {x}{a^2}+\frac {1}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)} \]
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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2320, 272, 46}
\begin {gather*} -\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}+\frac {x}{a^2}+\frac {1}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )^2} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {x}{a^2}+\frac {1}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 107, normalized size = 1.45 \begin {gather*} \frac {\frac {1}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n}+\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n}-\frac {\log \left (a^3 f g n \log (F)+a^2 b f \left (F^{g (e+f x)}\right )^n g n \log (F)\right )}{a^2 f g n}}{\log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 74, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{2}}-\frac {\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{2}}+\frac {1}{a \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}}{g f \ln \left (F \right ) n}\) | \(74\) |
default | \(\frac {\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{2}}-\frac {\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{2}}+\frac {1}{a \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}}{g f \ln \left (F \right ) n}\) | \(74\) |
risch | \(\frac {\ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right ) f g \,a^{2}}+\frac {1}{a f \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) g n \ln \left (F \right )}-\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}+\frac {a}{b}\right )}{\ln \left (F \right ) f g n \,a^{2}}\) | \(96\) |
norman | \(\frac {-\frac {b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{\ln \left (F \right ) f g n \,a^{2}}+\frac {b \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right ) {\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{a^{2} \ln \left (F \right ) f g}+\frac {\ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}{\ln \left (F \right ) a f g}}{a +b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}-\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}\right )}{\ln \left (F \right ) f g n \,a^{2}}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 97, normalized size = 1.31 \begin {gather*} \frac {f g n x + g n e}{a^{2} f g n} + \frac {1}{{\left (F^{f g n x + g n e} a b + a^{2}\right )} f g n \log \left (F\right )} - \frac {\log \left (F^{f g n x + g n e} b + a\right )}{a^{2} f g n \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 104, normalized size = 1.41 \begin {gather*} \frac {F^{f g n x + g n e} b f g n x \log \left (F\right ) + a f g n x \log \left (F\right ) - {\left (F^{f g n x + g n e} b + a\right )} \log \left (F^{f g n x + g n e} b + a\right ) + a}{F^{f g n x + g n e} a^{2} b f g n \log \left (F\right ) + a^{3} f g n \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 66, normalized size = 0.89 \begin {gather*} \frac {1}{a^{2} f g n \log {\left (F \right )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\left (F \right )}} + \frac {x}{a^{2}} - \frac {\log {\left (\frac {a}{b} + \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log {\left (F \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.27, size = 111, normalized size = 1.50 \begin {gather*} \frac {\log \left ({\left | F \right |}^{f g n x} {\left | F \right |}^{g n e}\right )}{a^{2} f g n \log \left (F\right )} - \frac {\log \left ({\left | F^{f g n x} F^{g n e} b + a \right |}\right )}{a^{2} f g n \log \left (F\right )} + \frac {1}{{\left (F^{f g n x} F^{g n e} b + a\right )} a f g n \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.45, size = 80, normalized size = 1.08 \begin {gather*} \frac {x}{a^2}+\frac {1}{a\,f\,g\,n\,\ln \left (F\right )\,\left (a+b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\right )}-\frac {\ln \left (a+b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\right )}{a^2\,f\,g\,n\,\ln \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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