Integrand size = 30, antiderivative size = 218 \[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=\frac {9 e^{\frac {a \sqrt {-(1+m)^2}}{1+m}} x^{1+m} \left (c x^2\right )^{\frac {1}{6} (-1-m)}}{16 \sqrt {-(1+m)^2}}-\frac {9 e^{\frac {a (1+m)}{\sqrt {-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac {1+m}{6}}}{32 \sqrt {-(1+m)^2}}+\frac {e^{\frac {3 a (1+m)}{\sqrt {-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac {1+m}{2}}}{16 \sqrt {-(1+m)^2}}-\frac {e^{-\frac {3 a (1+m)}{\sqrt {-(1+m)^2}}} (1+m) x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)} \log (x)}{8 \sqrt {-(1+m)^2}} \]
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Time = 0.33 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4581, 4577} \[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=\frac {9 e^{\frac {a \sqrt {-(m+1)^2}}{m+1}} x^{m+1} \left (c x^2\right )^{\frac {1}{6} (-m-1)}}{16 \sqrt {-(m+1)^2}}-\frac {9 e^{\frac {a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \left (c x^2\right )^{\frac {m+1}{6}}}{32 \sqrt {-(m+1)^2}}+\frac {e^{\frac {3 a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \left (c x^2\right )^{\frac {m+1}{2}}}{16 \sqrt {-(m+1)^2}}-\frac {(m+1) e^{-\frac {3 a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \log (x) \left (c x^2\right )^{\frac {1}{2} (-m-1)}}{8 \sqrt {-(m+1)^2}} \]
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Rule 4577
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \left (x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{2}} \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log (x)\right ) \, dx,x,c x^2\right ) \\ & = \frac {\left (\sqrt {-(1+m)^2} x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \text {Subst}\left (\int \left (\frac {e^{-\frac {3 a (1+m)}{\sqrt {-(1+m)^2}}}}{x}-3 e^{\frac {a \sqrt {-(1+m)^2}}{1+m}} x^{\frac {1}{3} (-2+m)}-e^{\frac {3 a (1+m)}{\sqrt {-(1+m)^2}}} x^m+3 e^{\frac {a (1+m)}{\sqrt {-(1+m)^2}}} x^{\frac {1}{3} (-1+2 m)}\right ) \, dx,x,c x^2\right )}{16 (1+m)} \\ & = \frac {9 e^{\frac {a \sqrt {-(1+m)^2}}{1+m}} x^{1+m} \left (c x^2\right )^{\frac {1}{6} (-1-m)}}{16 \sqrt {-(1+m)^2}}-\frac {9 e^{\frac {a (1+m)}{\sqrt {-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac {1+m}{6}}}{32 \sqrt {-(1+m)^2}}+\frac {e^{\frac {3 a (1+m)}{\sqrt {-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac {1+m}{2}}}{16 \sqrt {-(1+m)^2}}+\frac {e^{-\frac {3 a (1+m)}{\sqrt {-(1+m)^2}}} \sqrt {-(1+m)^2} x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)} \log (x)}{8 (1+m)} \\ \end{align*}
\[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=\int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx \]
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\[\int x^{m} {\sin \left (a +\frac {\ln \left (c \,x^{2}\right ) \sqrt {-\left (1+m \right )^{2}}}{6}\right )}^{3}d x\]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.45 \[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=-\frac {{\left (4 \, {\left (-i \, m - i\right )} e^{\left (-{\left (m + 1\right )} \log \left (c\right ) - 2 \, {\left (m + 1\right )} \log \left (x\right ) + 6 i \, a\right )} \log \left (x\right ) - 9 i \, e^{\left (-\frac {1}{3} \, {\left (m + 1\right )} \log \left (c\right ) - \frac {2}{3} \, {\left (m + 1\right )} \log \left (x\right ) + 2 i \, a\right )} + 18 i \, e^{\left (-\frac {2}{3} \, {\left (m + 1\right )} \log \left (c\right ) - \frac {4}{3} \, {\left (m + 1\right )} \log \left (x\right ) + 4 i \, a\right )} + 2 i\right )} e^{\left (\frac {1}{2} \, {\left (m + 1\right )} \log \left (c\right ) + 2 \, {\left (m + 1\right )} \log \left (x\right ) - 3 i \, a\right )}}{32 \, {\left (m + 1\right )}} \]
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\[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=\int x^{m} \sin ^{3}{\left (a + \frac {\sqrt {- m^{2} - 2 m - 1} \log {\left (c x^{2} \right )}}{6} \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.94 \[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=\frac {9 \, {\left (\cos \left (2 \, a\right ) \sin \left (3 \, a\right ) - \cos \left (3 \, a\right ) \sin \left (2 \, a\right )\right )} c^{\frac {5}{6} \, m + \frac {5}{6}} x^{\frac {5}{3}} x^{\frac {4}{3} \, m} + 18 \, {\left (\cos \left (3 \, a\right ) \sin \left (4 \, a\right ) - \cos \left (4 \, a\right ) \sin \left (3 \, a\right )\right )} c^{\frac {1}{2} \, m + \frac {1}{2}} x x^{\frac {2}{3} \, m} - 2 \, {\left (c^{\frac {7}{6} \, m + 1} x^{2} x^{2 \, m} \sin \left (3 \, a\right ) + 2 \, {\left ({\left (\cos \left (3 \, a\right )^{2} \sin \left (3 \, a\right ) + \sin \left (3 \, a\right )^{3}\right )} c^{\frac {1}{6} \, m} m + {\left (\cos \left (3 \, a\right )^{2} \sin \left (3 \, a\right ) + \sin \left (3 \, a\right )^{3}\right )} c^{\frac {1}{6} \, m}\right )} \log \left (x\right )\right )} c^{\frac {1}{6}} x^{\frac {1}{3}}}{32 \, {\left ({\left (\cos \left (3 \, a\right )^{2} + \sin \left (3 \, a\right )^{2}\right )} c^{\frac {2}{3} \, m} m + {\left (\cos \left (3 \, a\right )^{2} + \sin \left (3 \, a\right )^{2}\right )} c^{\frac {2}{3} \, m}\right )} c^{\frac {2}{3}} x^{\frac {1}{3}}} \]
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Result contains complex when optimal does not.
Time = 2.86 (sec) , antiderivative size = 1297, normalized size of antiderivative = 5.95 \[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=\text {Too large to display} \]
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Time = 29.61 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.33 \[ \int x^m \sin ^3\left (a+\frac {1}{6} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx=-\frac {\frac {1}{c^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}}\,x\,x^m\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (x^2\right )}^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}}\,1{}\mathrm {i}}{8\,m+8-\sqrt {-{\left (m+1\right )}^2}\,8{}\mathrm {i}}+\frac {c^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}\,x\,x^m\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (x^2\right )}^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{8\,m+8+\sqrt {-{\left (m+1\right )}^2}\,8{}\mathrm {i}}-\frac {\frac {1}{c^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{6}}}\,x\,x^m\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (x^2\right )}^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{6}}}\,\left (27\,m+27+\sqrt {-{\left (m+1\right )}^2}\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,{\left (m\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2}+\frac {c^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{6}}\,x\,x^m\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (x^2\right )}^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{6}}\,\left (27\,m+27-\sqrt {-{\left (m+1\right )}^2}\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,{\left (m\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2} \]
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