Optimal. Leaf size=22 \[ -\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2302, 30} \[ -\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=-\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ -\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 62, normalized size = 2.82 \[ -\frac {1}{2 \, {\left (b^{3} n^{3} \log \relax (x)^{2} + b^{3} n \log \relax (c)^{2} + 2 \, a b^{2} n \log \relax (c) + a^{2} b n + 2 \, {\left (b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} \log \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 21, normalized size = 0.95 \[ -\frac {1}{2 \, {\left (b n \log \relax (x) + b \log \relax (c) + a\right )}^{2} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 0.95 \[ -\frac {1}{2 \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 39, normalized size = 1.77 \[ -\frac {1}{2\,n\,a^2\,b+4\,n\,a\,b^2\,\ln \left (c\,x^n\right )+2\,n\,b^3\,{\ln \left (c\,x^n\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 93.79, size = 121, normalized size = 5.50 \[ \begin {cases} \frac {\tilde {\infty } \log {\relax (x )}}{\log {\relax (c )}^{3}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\tilde {\infty } n \log {\relax (x )} & \text {for}\: a = - b \left (n \log {\relax (x )} + \log {\relax (c )}\right ) \\\frac {\log {\relax (x )}}{a^{3}} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{\left (a + b \log {\relax (c )}\right )^{3}} & \text {for}\: n = 0 \\- \frac {1}{2 a^{2} b n + 4 a b^{2} n^{2} \log {\relax (x )} + 4 a b^{2} n \log {\relax (c )} + 2 b^{3} n^{3} \log {\relax (x )}^{2} + 4 b^{3} n^{2} \log {\relax (c )} \log {\relax (x )} + 2 b^{3} n \log {\relax (c )}^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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