Optimal. Leaf size=33 \[ \frac{5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac{20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]
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Rubi [A] time = 0.0608089, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {12, 444, 43} \[ \frac{5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac{20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx &=\operatorname{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 \operatorname{Subst}\left (\int x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int (2-3 x)^{3/5} (1-2 x) \, dx,x,\sin ^2(x)\right )\\ &=2 \operatorname{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*}
Mathematica [A] time = 0.0413974, size = 29, normalized size = 0.88 \[ -\frac{5 (3 \cos (2 x)+1)^{8/5} (24 \cos (2 x)-5)}{936\ 2^{3/5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 38, normalized size = 1.2 \begin{align*}{\frac{5}{12} \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{{\frac{8}{5}}}}-{\frac{20}{117} \left ({\frac{1}{2}}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) ^{{\frac{13}{5}}}}-{\frac{5}{18} \left ({\frac{1}{2}}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) ^{{\frac{8}{5}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969511, size = 34, normalized size = 1.03 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.46549, size = 89, normalized size = 2.7 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac{3}{5}} \sin \left (4 \, x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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