Optimal. Leaf size=96 \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e n^2}-\frac{d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Rubi [A] time = 0.063036, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2297, 2300, 2178} \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e n^2}-\frac{d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2297
Rule 2300
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{e}\\ &=-\frac{d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e n}\\ &=-\frac{d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e n^2}\\ &=\frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e n^2}-\frac{d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.049329, size = 123, normalized size = 1.28 \[ -\frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (b n e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}}-\left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )\right )}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.081, size = 457, normalized size = 4.8 \begin{align*} -2\,{\frac{ex+d}{ \left ( 2\,a+2\,b\ln \left ( c \right ) +2\,b\ln \left ( \left ( ex+d \right ) ^{n} \right ) -ib\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) +ib\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}+ib\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}-ib\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3} \right ) bne}}-{\frac{1}{{b}^{2}{n}^{2}e}{\it Ei} \left ( 1,-\ln \left ( ex+d \right ) -{\frac{-ib\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) +ib\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}+ib\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}-ib\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}+2\,b\ln \left ( c \right ) +2\,b \left ( \ln \left ( \left ( ex+d \right ) ^{n} \right ) -n\ln \left ( ex+d \right ) \right ) +2\,a}{2\,bn}} \right ){{\rm e}^{{\frac{ib\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) -ib\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}+ib\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}+2\,bn\ln \left ( ex+d \right ) -2\,b\ln \left ( c \right ) -2\,b\ln \left ( \left ( ex+d \right ) ^{n} \right ) -2\,a}{2\,bn}}}}} \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{e x + d}{b^{2} e n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} e n \log \left (c\right ) + a b e n} + \int \frac{1}{b^{2} n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} n \log \left (c\right ) + a b n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17017, size = 292, normalized size = 3.04 \begin{align*} -\frac{{\left ({\left (b e n x + b d n\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} -{\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )} \logintegral \left ({\left (e x + d\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )}}{b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e 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 + e x\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 = 1.28831, size = 414, normalized size = 4.31 \begin{align*} \frac{b n{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n}\right )} \log \left (x e + d\right )}{{\left (b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \left (c\right ) + a b^{2} n^{2} e\right )} c^{\left (\frac{1}{n}\right )}} - \frac{{\left (x e + d\right )} b n}{b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \left (c\right ) + a b^{2} n^{2} e} + \frac{b{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \left (c\right ) + a b^{2} n^{2} e\right )} c^{\left (\frac{1}{n}\right )}} + \frac{a{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n}\right )}}{{\left (b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \left (c\right ) + a b^{2} n^{2} e\right )} c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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