Optimal. Leaf size=123 \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b^2 f m^2 n^2}-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]
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Rubi [A] time = 0.167533, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2297, 2300, 2178, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b^2 f m^2 n^2}-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2297
Rule 2300
Rule 2178
Rule 2445
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^2} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^2} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}+\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^n x^{m n}\right )} \, dx,x,e+f x\right )}{b f m n},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}+\operatorname{Subst}\left (\frac{\left ((e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{m n}}}{a+b x} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{b f m^2 n^2},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{e^{-\frac{a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b^2 f m^2 n^2}-\frac{e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.110355, size = 163, normalized size = 1.33 \[ -\frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \left (b m n e^{\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac{1}{m n}}-\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )\right )}{b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{-2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{f x + e}{b^{2} f m n \log \left ({\left ({\left (f x + e\right )}^{m}\right )}^{n}\right ) + a b f m n +{\left (f m n \log \left (c\right ) + f m n \log \left (d^{n}\right )\right )} b^{2}} + \int \frac{1}{b^{2} m n \log \left ({\left ({\left (f x + e\right )}^{m}\right )}^{n}\right ) + a b m n +{\left (m n \log \left (c\right ) + m n \log \left (d^{n}\right )\right )} b^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25319, size = 425, normalized size = 3.46 \begin{align*} -\frac{{\left ({\left (b f m n x + b e m n\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} -{\left (b m n \log \left (f x + e\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )} \logintegral \left ({\left (f x + e\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}\right )\right )} e^{\left (-\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}}{b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24344, size = 801, normalized size = 6.51 \begin{align*} -\frac{{\left (f x + e\right )} b m n}{b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}} + \frac{b m n{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )} \log \left (f x + e\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} + \frac{b n{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )} \log \left (d\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} + \frac{b{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )} \log \left (c\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} + \frac{a{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )}}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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