Optimal. Leaf size=111 \[ \frac {x}{a^3}+\frac {1}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {1}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f g n \log (F)} \]
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Rubi [A]
time = 0.05, antiderivative size = 111, 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^3 f g n \log (F)}+\frac {x}{a^3}+\frac {1}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {1}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \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 )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )^3} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^3} \, 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^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {x}{a^3}+\frac {1}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {1}{a^2 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^3 f g n \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 128, normalized size = 1.15 \begin {gather*} \frac {\frac {3 a+2 b \left (F^{g (e+f x)}\right )^n}{2 a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n}+\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a^3 f g n}-\frac {\log \left (a^4 f g n \log (F)+a^3 b f \left (F^{g (e+f x)}\right )^n g n \log (F)\right )}{a^3 f g n}}{\log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 96, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{3}}-\frac {\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}+\frac {1}{2 a \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2}}}{g f \ln \left (F \right ) n}\) | \(96\) |
default | \(\frac {\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{3}}-\frac {\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}+\frac {1}{2 a \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2}}}{g f \ln \left (F \right ) n}\) | \(96\) |
risch | \(\frac {\ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right ) a^{3} f g}+\frac {2 b \left (F^{g \left (f x +e \right )}\right )^{n}+3 a}{2 \ln \left (F \right ) f g n \,a^{2} \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2}}-\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}+\frac {a}{b}\right )}{\ln \left (F \right ) a^{3} f g n}\) | \(115\) |
norman | \(\frac {\frac {b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{\ln \left (F \right ) f g n \,a^{2}}+\frac {b^{2} x \,{\mathrm e}^{2 n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{a^{3}}+\frac {x}{a}+\frac {2 b x \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{a^{2}}+\frac {3}{2 \ln \left (F \right ) a f g n}}{\left (a +b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}\right )^{2}}-\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}\right )}{\ln \left (F \right ) a^{3} f g n}\) | \(161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 143, normalized size = 1.29 \begin {gather*} \frac {2 \, F^{f g n x + g n e} b + 3 \, a}{2 \, {\left (2 \, F^{f g n x + g n e} a^{3} b + F^{2 \, f g n x + 2 \, g n e} a^{2} b^{2} + a^{4}\right )} f g n \log \left (F\right )} + \frac {f g n x + g n e}{a^{3} f g n} - \frac {\log \left (F^{f g n x + g n e} b + a\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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Fricas [A]
time = 0.45, size = 197, normalized size = 1.77 \begin {gather*} \frac {2 \, F^{2 \, f g n x + 2 \, g n e} b^{2} f g n x \log \left (F\right ) + 2 \, a^{2} f g n x \log \left (F\right ) + 2 \, {\left (2 \, a b f g n x \log \left (F\right ) + a b\right )} F^{f g n x + g n e} + 3 \, a^{2} - 2 \, {\left (2 \, F^{f g n x + g n e} a b + F^{2 \, f g n x + 2 \, g n e} b^{2} + a^{2}\right )} \log \left (F^{f g n x + g n e} b + a\right )}{2 \, {\left (2 \, F^{f g n x + g n e} a^{4} b f g n \log \left (F\right ) + F^{2 \, f g n x + 2 \, g n e} a^{3} b^{2} f g n \log \left (F\right ) + a^{5} f g n \log \left (F\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 116, normalized size = 1.05 \begin {gather*} \frac {3 a + 2 b \left (F^{g \left (e + f x\right )}\right )^{n}}{2 a^{4} f g n \log {\left (F \right )} + 4 a^{3} b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\left (F \right )} + 2 a^{2} b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log {\left (F \right )}} + \frac {x}{a^{3}} - \frac {\log {\left (\frac {a}{b} + \left (F^{g \left (e + f x\right )}\right )^{n} \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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Giac [A]
time = 1.97, size = 136, normalized size = 1.23 \begin {gather*} \frac {\log \left ({\left | F \right |}^{f g n x} {\left | F \right |}^{g n e}\right )}{a^{3} 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^{3} f g n \log \left (F\right )} + \frac {2 \, F^{f g n x} F^{g n e} a b + 3 \, a^{2}}{2 \, {\left (F^{f g n x} F^{g n e} b + a\right )}^{2} 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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Mupad [B]
time = 3.50, size = 142, normalized size = 1.28 \begin {gather*} \frac {x}{a^3}+\frac {1}{2\,a\,f\,g\,n\,\ln \left (F\right )\,\left (a^2+b^2\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}+2\,a\,b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\right )}+\frac {1}{a^2\,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^3\,f\,g\,n\,\ln \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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