Integrand size = 17, antiderivative size = 149 \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3} n}+\frac {\log (x)}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n} \]
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Time = 0.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {327, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )}{6 a^{4/3} n}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )}{3 a^{4/3} n}+\frac {\log (x)}{a} \]
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Rule 31
Rule 206
Rule 210
Rule 327
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{b+a x^3} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\log (x)}{a}-\frac {b \text {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,\log \left (c x^n\right )\right )}{a n} \\ & = \frac {\log (x)}{a}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,\log \left (c x^n\right )\right )}{3 a n}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{3 a n} \\ & = \frac {\log (x)}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{6 a^{4/3} n}-\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{2 a n} \\ & = \frac {\log (x)}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}\right )}{a^{4/3} n} \\ & = \frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} n}+\frac {\log (x)}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\frac {2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}}{\sqrt {3}}\right )+6 \sqrt [3]{a} n \log (x)+\sqrt [3]{b} \left (-2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )+\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )\right )}{6 a^{4/3} n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.98 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 n^{3} a^{4} \textit {\_Z}^{3}+b \right )}{\sum }\textit {\_R} \ln \left (\ln \left (x^{n}\right )-3 a n \textit {\_R} -\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \left (c \right )\right )\right )\) | \(118\) |
default | \(\frac {\frac {\ln \left (c \,x^{n}\right )}{a}-\frac {\left (\frac {\ln \left (\ln \left (c \,x^{n}\right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (\ln \left (c \,x^{n}\right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} \ln \left (c \,x^{n}\right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \ln \left (c \,x^{n}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right ) b}{a}}{n}\) | \(132\) |
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Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\frac {6 \, n \log \left (x\right ) + 2 \, \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, {\left (\sqrt {3} a n \log \left (x\right ) + \sqrt {3} a \log \left (c\right )\right )} \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2} + {\left (n \log \left (x\right ) + \log \left (c\right )\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (n \log \left (x\right ) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \log \left (c\right )\right )}{6 \, a n} \]
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Time = 26.84 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.64 \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\begin {cases} \tilde {\infty } \log {\left (c \right )}^{3} \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\begin {cases} 0 & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \wedge \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{4}}{4 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {6 {G_{5, 5}^{5, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {6 {G_{5, 5}^{0, 5}\left (\begin {matrix} 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}}{b} & \text {for}\: a = 0 \\\frac {\log {\left (c \right )}^{3} \log {\left (x \right )}}{a \log {\left (c \right )}^{3} + b} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\sqrt [3]{- \frac {b}{a}} \log {\left (- \sqrt [3]{- \frac {b}{a}} + \log {\left (c x^{n} \right )} \right )}}{3 a n} - \frac {\sqrt [3]{- \frac {b}{a}} \log {\left (4 \left (- \frac {b}{a}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {b}{a}} \log {\left (c x^{n} \right )} + 4 \log {\left (c x^{n} \right )}^{2} \right )}}{6 a n} - \frac {\sqrt {3} \sqrt [3]{- \frac {b}{a}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \log {\left (c x^{n} \right )}}{3 \sqrt [3]{- \frac {b}{a}}} \right )}}{3 a n} + \frac {\log {\left (c x^{n} \right )}}{a n} & \text {otherwise} \end {cases} \]
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\[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\int { \frac {1}{a x + \frac {b x}{\log \left (c x^{n}\right )^{3}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (115) = 230\).
Time = 0.37 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.72 \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\frac {\log \left (x\right )}{a} + \frac {2 \, \sqrt {3} \left (-\frac {b n^{6}}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, a n \log \left (x\right ) - 2 \, a \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac {1}{3}}}{2 \, \sqrt {3} a n \log \left (x\right ) + \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt {3} a \log \left ({\left | c \right |}\right ) + 2 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}}}\right ) + \left (-\frac {b n^{6}}{a}\right )^{\frac {1}{3}} \log \left (\frac {1}{4} \, {\left (\pi a n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}^{2}\right ) - \left (-\frac {b n^{6}}{a}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )} - 2 \, a n \log \left (x\right ) - 2 \, a \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac {1}{3}}\right )}^{2} + {\left (2 \, \sqrt {3} a n \log \left (x\right ) + \pi a {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt {3} a \log \left ({\left | c \right |}\right ) + 2 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}}\right )}^{2}\right )}{6 \, a n^{3}} \]
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Time = 3.66 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.17 \[ \int \frac {1}{a x+\frac {b x}{\log ^3\left (c x^n\right )}} \, dx=\frac {\ln \left (x\right )}{a}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {3\,{\left (-b\right )}^{4/3}\,n}{a^{7/3}\,x^2}+\frac {3\,b\,n\,\ln \left (c\,x^n\right )}{a^2\,x^2}\right )}{3\,a^{4/3}\,n}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {3\,b\,n\,\ln \left (c\,x^n\right )}{a^2\,x^2}+\frac {3\,{\left (-b\right )}^{4/3}\,n\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{7/3}\,x^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,n}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {3\,b\,n\,\ln \left (c\,x^n\right )}{a^2\,x^2}-\frac {3\,{\left (-b\right )}^{4/3}\,n\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{7/3}\,x^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,n} \]
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