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 = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2178
Rule 2297
Rule 2300
Rule 2389
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.44, size = 117, normalized size = 1.22 \[ -\frac {{\left ({\left (b e n x + b d n\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} - {\left (b n \log \left (e x + d\right ) + b \log \relax (c) + a\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )}}{b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \relax (c) + a b^{2} e n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 307, normalized size = 3.20 \[ \frac {b n {\rm Ei}\left (\frac {\log \relax (c)}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (x e + d\right )}{{\left (b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \relax (c) + a b^{2} n^{2} e\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (x e + d\right )} b n}{b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \relax (c) + a b^{2} n^{2} e} + \frac {b {\rm Ei}\left (\frac {\log \relax (c)}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \relax (c)}{{\left (b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \relax (c) + a b^{2} n^{2} e\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a {\rm Ei}\left (\frac {\log \relax (c)}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{{\left (b^{3} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \relax (c) + a b^{2} n^{2} e\right )} c^{\left (\frac {1}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 456, normalized size = 4.75 \[ -\frac {\left (e x +d \right ) c^{-\frac {1}{n}} \left (\left (e x +d \right )^{n}\right )^{-\frac {1}{n}} \Ei \left (1, -\ln \left (e x +d \right )-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \relax (c )+2 a +2 \left (-n \ln \left (e x +d \right )+\ln \left (\left (e x +d \right )^{n}\right )\right ) b}{2 b n}\right ) {\mathrm e}^{-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 a}{2 b n}}}{b^{2} e \,n^{2}}-\frac {2 \left (e x +d \right )}{\left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \relax (c )+2 b \ln \left (\left (e x +d \right )^{n}\right )+2 a \right ) b e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {e x + d}{b^{2} e n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} e n \log \relax (c) + a b e n} + \int \frac {1}{b^{2} n \log \left ({\left (e x + d\right )}^{n}\right ) + b^{2} n \log \relax (c) + a b n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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