Optimal. Leaf size=102 \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2} \]
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 )^3} \, dx &=-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}-\frac {\int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 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}{2 b^2 n^2}\\ &=-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 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 )}{2 b^2 n^3 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 )}{2 b^3 n^3 x}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 94, normalized size = 0.92 \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right )^2 \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )+b n \left (a+b \log \left (c x^n\right )-b n\right )}{2 b^3 n^3 x \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 192, normalized size = 1.88 \[ \frac {b^{2} n^{2} \log \relax (x) - b^{2} n^{2} + b^{2} n \log \relax (c) + a b n + {\left (b^{2} n^{2} x \log \relax (x)^{2} + b^{2} x \log \relax (c)^{2} + 2 \, a b x \log \relax (c) + a^{2} x + 2 \, {\left (b^{2} n x \log \relax (c) + a b n x\right )} \log \relax (x)\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )}}{x}\right )}{2 \, {\left (b^{5} n^{5} x \log \relax (x)^{2} + b^{5} n^{3} x \log \relax (c)^{2} + 2 \, a b^{4} n^{3} x \log \relax (c) + a^{2} b^{3} n^{3} x + 2 \, {\left (b^{5} n^{4} x \log \relax (c) + a b^{4} n^{4} x\right )} \log \relax (x)\right )}} \]
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 \[ \int \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.95, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x^{2}}\, dx \]
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 \[ -\frac {b {\left (n - \log \relax (c)\right )} - b \log \left (x^{n}\right ) - a}{2 \, {\left (b^{4} n^{2} x \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{4} n^{2} \log \relax (c) + a b^{3} n^{2}\right )} x \log \left (x^{n}\right ) + {\left (b^{4} n^{2} \log \relax (c)^{2} + 2 \, a b^{3} n^{2} \log \relax (c) + a^{2} b^{2} n^{2}\right )} x\right )}} + \int \frac {1}{2 \, {\left (b^{3} n^{2} x^{2} \log \left (x^{n}\right ) + {\left (b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} x^{2}\right )}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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