Integrand size = 17, antiderivative size = 33 \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=\frac {5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac {20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {12, 455, 45} \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=\frac {5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac {20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]
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Rule 12
Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int 4 x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right ) \\ & = 4 \text {Subst}\left (\int x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right ) \\ & = 2 \text {Subst}\left (\int (2-3 x)^{3/5} (1-2 x) \, dx,x,\sin ^2(x)\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {1}{3} (2-3 x)^{3/5}+\frac {2}{3} (2-3 x)^{8/5}\right ) \, dx,x,\sin ^2(x)\right ) \\ & = \frac {5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac {20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=-\frac {5 (1+3 \cos (2 x))^{8/5} (-5+24 \cos (2 x))}{936\ 2^{3/5}} \]
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Time = 0.47 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {5 {\left (2-3 \left (\sin ^{2}\left (x \right )\right )\right )}^{\frac {8}{5}}}{12}-\frac {5 \left (16+48 \cos \left (2 x \right )\right )^{\frac {13}{5}}}{239616}-\frac {5 \left (16+48 \cos \left (2 x \right )\right )^{\frac {8}{5}}}{4608}\) | \(38\) |
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=-\frac {5}{468} \, {\left (144 \, \cos \left (x\right )^{4} - 135 \, \cos \left (x\right )^{2} + 29\right )} {\left (3 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {3}{5}} \]
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Timed out. \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=-\frac {20}{117} \, {\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac {13}{5}} + \frac {5}{36} \, {\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac {8}{5}} \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=-\frac {20}{117} \, {\left (3 \, \sin \left (x\right )^{2} - 2\right )}^{2} {\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac {3}{5}} + \frac {5}{36} \, {\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac {8}{5}} \]
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Timed out. \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx=\int \sin \left (4\,x\right )\,{\left (2-3\,{\sin \left (x\right )}^2\right )}^{3/5} \,d x \]
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