Optimal. Leaf size=144 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n} \]
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Rubi [A] time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{a x+b x \log ^3\left (c x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\log \left (c x^n\right )\right )}{3 a^{2/3} n}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{3 a^{2/3} n}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{2 \sqrt [3]{a} n}-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b} n}\\ &=-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} n}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )}{3 a^{2/3} \sqrt [3]{b} n}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{2/3} \sqrt [3]{b} n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 112, normalized size = 0.78 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3} \log ^2\left (c x^n\right )\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \log \left (c x^n\right )\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \log \left (c x^n\right )}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{6 a^{2/3} \sqrt [3]{b} n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 480, normalized size = 3.33 \[ \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b n^{3} \log \relax (x)^{3} + 6 \, a b n^{2} \log \relax (c) \log \relax (x)^{2} + 6 \, a b n \log \relax (c)^{2} \log \relax (x) + 2 \, a b \log \relax (c)^{3} - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b n^{2} \log \relax (x)^{2} + 4 \, a b n \log \relax (c) \log \relax (x) + 2 \, a b \log \relax (c)^{2} + \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \relax (x) + \log \relax (c)\right )} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} {\left (a n \log \relax (x) + a \log \relax (c)\right )}}{b n^{3} \log \relax (x)^{3} + 3 \, b n^{2} \log \relax (c) \log \relax (x)^{2} + 3 \, b n \log \relax (c)^{2} \log \relax (x) + b \log \relax (c)^{3} + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n^{2} \log \relax (x)^{2} + 2 \, a b n \log \relax (c) \log \relax (x) + a b \log \relax (c)^{2} - \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \relax (x) + \log \relax (c)\right )} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n \log \relax (x) + a b \log \relax (c) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b n}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \relax (x) + \log \relax (c)\right )} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n^{2} \log \relax (x)^{2} + 2 \, a b n \log \relax (c) \log \relax (x) + a b \log \relax (c)^{2} - \left (a^{2} b\right )^{\frac {2}{3}} {\left (n \log \relax (x) + \log \relax (c)\right )} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b n \log \relax (x) + a b \log \relax (c) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 239, normalized size = 1.66 \[ \frac {1}{3} \, \sqrt {3} \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )} - 2 \, b n \log \relax (x) - 2 \, b \log \left ({\left | c \right |}\right ) - 2 \, \left (a b^{2}\right )^{\frac {1}{3}}}{2 \, \sqrt {3} b n \log \relax (x) + \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )} + 2 \, \sqrt {3} b \log \left ({\left | c \right |}\right ) - 2 \, \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right ) + \frac {1}{6} \, \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) - \frac {1}{6} \, \left (\frac {1}{a^{2} b n^{3}}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )} - 2 \, b n \log \relax (x) - 2 \, b \log \left ({\left | c \right |}\right ) - 2 \, \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + {\left (2 \, \sqrt {3} b n \log \relax (x) + \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )} + 2 \, \sqrt {3} b \log \left ({\left | c \right |}\right ) - 2 \, \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 120, normalized size = 0.83 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b n}+\frac {\ln \left (\ln \left (c \,x^{n}\right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b n}-\frac {\ln \left (\ln \left (c \,x^{n}\right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \ln \left (c \,x^{n}\right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b n} \]
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 \[ \int \frac {1}{b x \log \left (c x^{n}\right )^{3} + a x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.28, size = 153, normalized size = 1.06 \[ \frac {\ln \left (\frac {3\,a^{1/3}\,n}{b^{4/3}\,x^2}+\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}\right )}{3\,a^{2/3}\,b^{1/3}\,n}+\frac {\ln \left (\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}+\frac {3\,a^{1/3}\,n\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{4/3}\,x^2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,n}-\frac {\ln \left (\frac {3\,n\,\ln \left (c\,x^n\right )}{b\,x^2}-\frac {3\,a^{1/3}\,n\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{4/3}\,x^2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 52.76, size = 330, normalized size = 2.29 \[ \begin {cases} \frac {\tilde {\infty } \log {\relax (x )}}{\log {\relax (c )}^{3}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b \log {\relax (c )}^{3}} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\- \frac {1}{b \left (2 n^{3} \log {\relax (x )}^{2} + 4 n^{2} \log {\relax (c )} \log {\relax (x )} + 2 n \log {\relax (c )}^{2}\right )} & \text {for}\: a = 0 \\- \frac {\sqrt [3]{-1} \sqrt [3]{\frac {1}{b}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{3 a^{\frac {2}{3}} n} + \frac {\sqrt [3]{-1} \sqrt [3]{\frac {1}{b}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} n \sqrt [3]{\frac {1}{b}} \log {\relax (x )} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} \log {\relax (c )} + 4 n^{2} \log {\relax (x )}^{2} + 8 n \log {\relax (c )} \log {\relax (x )} + 4 \log {\relax (c )}^{2} \right )}}{6 a^{\frac {2}{3}} n} - \frac {\sqrt [3]{-1} \sqrt {3} \sqrt [3]{\frac {1}{b}} \operatorname {atan}{\left (- \frac {\sqrt {3}}{3} + \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} n \log {\relax (x )}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} + \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \log {\relax (c )}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )}}{3 a^{\frac {2}{3}} n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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