Integrand size = 28, antiderivative size = 178 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{8 x^2} \]
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Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4581, 4577} \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {\sqrt {-\frac {1}{n^2}} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 \sqrt {-\frac {1}{n^2}} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 \sqrt {-\frac {1}{n^2}} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {\sqrt {-\frac {1}{n^2}} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{8 x^2} \]
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Rule 4577
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x^2} \\ & = -\frac {\left (\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{2/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 {8}{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^{-\frac {4+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x^2} \\ & = -\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{8 x^2} \\ \end{align*}
\[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \]
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Time = 88.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \(\frac {-47 n \left (n +\frac {40 \ln \left (c \,x^{n}\right )}{47}\right ) \sqrt {-\frac {1}{n^{2}}}\, \cos \left (2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}+3 a \right )+\left (-27 n -40 \ln \left (c \,x^{n}\right )\right ) \sin \left (2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}+3 a \right )-45 n \left (\cos \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {2}{3}}\right )\right ) n \sqrt {-\frac {1}{n^{2}}}+3 \sin \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {2}{3}}\right )\right )\right )}{320 x^{2} n}\) | \(136\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {{\left (-24 i \, x^{4} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {8}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac {4}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {3 i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{64 \, x^{4}} \]
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Time = 54.03 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=- \frac {9 n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{64 x^{2}} - \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x^{2}} - \frac {27 \sin {\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{64 x^{2}} + \frac {\sin {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{16 x^{2}} - \frac {\log {\left (c x^{n} \right )} \sin {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 n x^{2}} \]
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none
Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {{\left (8 \, c^{\frac {14}{3 \, n}} x^{2} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) + 9 \, c^{\frac {2}{n}} x^{4} \sin \left (a\right ) - 2 \, c^{\frac {2}{3 \, n}} x^{2} {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \sin \left (3 \, a\right ) + 18 \, c^{\frac {10}{3 \, n}} e^{\left (\frac {4 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {2 \, \log \left (x^{n}\right )}{3 \, n} - 4 \, \log \left (x\right )\right )}}{64 \, c^{\frac {8}{3 \, n}} x^{2}} \]
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\[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (\frac {2}{3} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{3}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+\frac {2\,\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{3}\right )}^3}{x^3} \,d x \]
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