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 = 0.05, 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 = {2436, 2334,
2337, 2209} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2334
Rule 2337
Rule 2436
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\frac {\text {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 {\text {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 ) \text {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 \begin {gather*} -\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (b e^{\frac {a}{b n}} n \left (c (d+e x)^n\right )^{\frac {1}{n}}-\text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.68, size = 456, normalized size = 4.75
method | result | size |
risch | \(-\frac {2 \left (e x +d \right )}{\left (2 a +2 b \ln \left (c \right )+2 b \ln \left (\left (e x +d \right )^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}\right ) b n e}-\frac {\left (e x +d \right ) c^{-\frac {1}{n}} \left (\left (e x +d \right )^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 a}{2 b n}} \expIntegral \left (1, -\ln \left (e x +d \right )-\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (\left (e x +d \right )^{n}\right )-n \ln \left (e x +d \right )\right )+2 a}{2 b n}\right )}{b^{2} n^{2} e}\) | \(456\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 124, normalized size = 1.29 \begin {gather*} -\frac {{\left ({\left (b n x e + b d n\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b n \log \left (x e + d\right ) + b \log \left (c\right ) + a\right )} \operatorname {log\_integral}\left ({\left (x e + 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} n^{3} e \log \left (x e + d\right ) + b^{3} n^{2} e \log \left (c\right ) + a b^{2} n^{2} e} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 307 vs.
\(2 (98) = 196\).
time = 2.72, size = 307, normalized size = 3.20 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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