Optimal. Leaf size=98 \[ \frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3}-\frac {x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2297, 2300, 2178} \[ \frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3}-\frac {x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2297
Rule 2300
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx &=-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {\int \frac {1}{a+b \log \left (c x^n\right )} \, dx}{2 b^2 n^2}\\ &=-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b^2 n^3}\\ &=\frac {e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3}-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 0.84 \[ \frac {x \left (e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )-\frac {b n \left (a+b \log \left (c x^n\right )+b n\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 198, normalized size = 2.02 \[ -\frac {{\left ({\left (b^{2} n^{2} x \log \relax (x) + b^{2} n x \log \relax (c) + {\left (b^{2} n^{2} + a b n\right )} x\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} - {\left (b^{2} n^{2} \log \relax (x)^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + a^{2} + 2 \, {\left (b^{2} n \log \relax (c) + a b n\right )} \log \relax (x)\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {b \log \relax (c) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )}}{2 \, {\left (b^{5} n^{5} \log \relax (x)^{2} + b^{5} n^{3} \log \relax (c)^{2} + 2 \, a b^{4} n^{3} \log \relax (c) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \relax (c) + a b^{4} n^{4}\right )} \log \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 982, normalized size = 10.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 459, normalized size = 4.68 \[ -\frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} \Ei \left (1, -\ln \relax (x )-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \relax (c )+2 a +2 \left (-n \ln \relax (x )+\ln \left (x^{n}\right )\right ) b}{2 b n}\right ) {\mathrm e}^{-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{2 b n}}}{2 b^{3} n^{3}}-\frac {-i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b n x +2 b x \ln \relax (c )+2 b x \ln \left (x^{n}\right )+2 a x}{\left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \relax (c )+2 b \ln \left (x^{n}\right )+2 a \right )^{2} b^{2} n^{2}} \]
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 \[ -\frac {b x \log \left (x^{n}\right ) + {\left (b {\left (n + \log \relax (c)\right )} + a\right )} x}{2 \, {\left (b^{4} n^{2} \log \relax (c)^{2} + b^{4} n^{2} \log \left (x^{n}\right )^{2} + 2 \, a b^{3} n^{2} \log \relax (c) + a^{2} b^{2} n^{2} + 2 \, {\left (b^{4} n^{2} \log \relax (c) + a b^{3} n^{2}\right )} \log \left (x^{n}\right )\right )}} + \int \frac {1}{2 \, {\left (b^{3} n^{2} \log \relax (c) + b^{3} n^{2} \log \left (x^{n}\right ) + a b^{2} n^{2}\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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 \[ \int \frac {1}{\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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