Integrand size = 19, antiderivative size = 18 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {1}{7} \left (a^2-4 \cos ^2(x)\right )^{7/4} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 267} \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {1}{7} \left (a^2+4 \sin ^2(x)-4\right )^{7/4} \]
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Rule 12
Rule 267
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int 2 x \left (-4+a^2+4 x^2\right )^{3/4} \, dx,x,\sin (x)\right ) \\ & = 2 \text {Subst}\left (\int x \left (-4+a^2+4 x^2\right )^{3/4} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{7} \left (-4+a^2+4 \sin ^2(x)\right )^{7/4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {1}{7} \left (-4+a^2+4 \sin ^2(x)\right )^{7/4} \]
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Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\left (a^{2}-4 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {7}{4}}}{7}\) | \(15\) |
default | \(\frac {\left (a^{2}-4 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {7}{4}}}{7}\) | \(15\) |
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Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {1}{7} \, {\left (a^{2} - 4 \, \cos \left (x\right )^{2}\right )}^{\frac {7}{4}} \]
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Timed out. \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {1}{7} \, {\left (a^{2} - 4 \, \cos \left (x\right )^{2}\right )}^{\frac {7}{4}} \]
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Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {1}{7} \, {\left (a^{2} - 4 \, \cos \left (x\right )^{2}\right )}^{\frac {7}{4}} \]
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Time = 0.42 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx=\frac {{\left (a^2-4\,{\cos \left (x\right )}^2\right )}^{7/4}}{7} \]
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