Integrand size = 24, antiderivative size = 128 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9}{16} e^{a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\left .\frac {1}{3}\right /n}+\frac {1}{16} e^{-3 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \]
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Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4572, 4578} \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9}{16} x e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} x e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {1}{3}\right /n}+\frac {1}{16} x e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} x e^{3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]
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Rule 4572
Rule 4578
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n}}{x}+3 e^{a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{3 n}}+3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{3 n}}+e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{n}}\right ) \, dx,x,c x^n\right )}{8 n} \\ & = \frac {9}{16} e^{a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\left .\frac {1}{3}\right /n}+\frac {1}{16} e^{-3 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \\ \end{align*}
\[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
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\[\int {\cos \left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{3}\right )}^{3}d x\]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.66 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{32} \, {\left (9 \, x^{\frac {4}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 2 \, x^{2} + 12 \, e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, x^{\frac {2}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )}\right )} e^{\left (-\frac {3 i \, a n - \log \left (c\right )}{n}\right )} \]
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\[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \cos ^{3}{\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9 \, c^{\frac {5}{3 \, n}} x {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \cos \left (a\right ) + 4 \, c^{\frac {1}{3 \, n}} {\left (x^{n}\right )}^{\frac {1}{3 \, n}} \cos \left (3 \, a\right ) \log \left (x\right ) + 18 \, c^{\left (\frac {1}{n}\right )} x \cos \left (a\right ) + 2 \, c^{\frac {7}{3 \, n}} \cos \left (3 \, a\right ) e^{\left (\frac {\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac {4}{3 \, n}} {\left (x^{n}\right )}^{\frac {1}{3 \, n}}} \]
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Exception generated. \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: NotImplementedError} \]
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Time = 27.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.23 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}{3}}}\,\left (\frac {27}{64}+\frac {n\,\sqrt {-\frac {1}{n^2}}\,9{}\mathrm {i}}{64}\right )-x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}{3}}\,\left (-\frac {27}{64}+\frac {n\,\sqrt {-\frac {1}{n^2}}\,9{}\mathrm {i}}{64}\right )+\frac {x\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}+8{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}-8{}\mathrm {i}} \]
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