Optimal. Leaf size=48 \[ \frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]
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Rubi [A] time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2300, 2178} \[ \frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2300
Rubi steps
\begin {align*} \int \frac {1}{a+b \log \left (c x^n\right )} \, dx &=\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 )}{n}\\ &=\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 )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 1.00 \[ \frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 39, normalized size = 0.81 \[ \frac {e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {b \log \relax (c) + a}{b n}\right )}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 42, normalized size = 0.88 \[ \frac {{\rm Ei}\left (\frac {\log \relax (c)}{n} + \frac {a}{b n} + \log \relax (x)\right ) e^{\left (-\frac {a}{b n}\right )}}{b c^{\left (\frac {1}{n}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 240, normalized size = 5.00 \[ -\frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} \Ei \left (1, -\ln \relax (x )-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \relax (c )+2 a +2 \left (-n \ln \relax (x )+\ln \left (x^{n}\right )\right ) b}{2 b n}\right ) {\mathrm e}^{-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{2 b n}}}{b n} \]
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 \[ \int \frac {1}{b \log \left (c x^{n}\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{a+b\,\ln \left (c\,x^n\right )} \,d x \]
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 \[ \int \frac {1}{a + b \log {\left (c x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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