Optimal. Leaf size=73 \[ -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n}\\ &=-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int \frac {e^{-\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2 x}\\ &=-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 76, normalized size = 1.04 \[ -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right ) \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )+b n}{b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 88, normalized size = 1.21 \[ -\frac {{\left (b n x \log \relax (x) + b x \log \relax (c) + a x\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )}}{x}\right ) + b n}{b^{3} n^{3} x \log \relax (x) + b^{3} n^{2} x \log \relax (c) + a b^{2} n^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2}}\, dx \]
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 \[ -\frac {1}{b^{2} n x \log \left (x^{n}\right ) + {\left (b^{2} n \log \relax (c) + a b n\right )} x} - \int \frac {1}{b^{2} n x^{2} \log \left (x^{n}\right ) + {\left (b^{2} n \log \relax (c) + a b n\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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