Integrand size = 32, antiderivative size = 55 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=\frac {3}{4 \sqrt {1+2 \sin (x)}}-\frac {4}{\sqrt [4]{1+2 \sin (x)}}-\frac {1}{2} \sqrt {1+2 \sin (x)}+\frac {1}{12} (1+2 \sin (x))^{3/2} \]
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Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4441, 14} \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=\frac {1}{12} (2 \sin (x)+1)^{3/2}-\frac {1}{2} \sqrt {2 \sin (x)+1}-\frac {4}{\sqrt [4]{2 \sin (x)+1}}+\frac {3}{4 \sqrt {2 \sin (x)+1}} \]
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Rule 14
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+x^2+2 \sqrt [4]{1+2 x}}{(1+2 x)^{3/2}} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-3+8 x-2 x^4+x^8}{x^3} \, dx,x,\sqrt [4]{1+2 \sin (x)}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {3}{x^3}+\frac {8}{x^2}-2 x+x^5\right ) \, dx,x,\sqrt [4]{1+2 \sin (x)}\right ) \\ & = \frac {3}{4 \sqrt {1+2 \sin (x)}}-\frac {4}{\sqrt [4]{1+2 \sin (x)}}-\frac {1}{2} \sqrt {1+2 \sin (x)}+\frac {1}{12} (1+2 \sin (x))^{3/2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=-\frac {-3+\cos (2 x)+4 \sin (x)+24 \sqrt [4]{1+2 \sin (x)}}{6 \sqrt {1+2 \sin (x)}} \]
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Time = 0.49 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(-\frac {4}{\left (1+2 \sin \left (x \right )\right )^{\frac {1}{4}}}+\frac {\left (1+2 \sin \left (x \right )\right )^{\frac {3}{2}}}{12}+\frac {3}{4 \sqrt {1+2 \sin \left (x \right )}}-\frac {\sqrt {1+2 \sin \left (x \right )}}{2}\) | \(42\) |
default | \(-\frac {4}{\left (1+2 \sin \left (x \right )\right )^{\frac {1}{4}}}+\frac {\left (1+2 \sin \left (x \right )\right )^{\frac {3}{2}}}{12}+\frac {3}{4 \sqrt {1+2 \sin \left (x \right )}}-\frac {\sqrt {1+2 \sin \left (x \right )}}{2}\) | \(42\) |
parts | \(-\frac {4}{\left (1+2 \sin \left (x \right )\right )^{\frac {1}{4}}}+\frac {\left (1+2 \sin \left (x \right )\right )^{\frac {3}{2}}}{12}+\frac {3}{4 \sqrt {1+2 \sin \left (x \right )}}-\frac {\sqrt {1+2 \sin \left (x \right )}}{2}\) | \(42\) |
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=-\frac {{\left (\cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 2\right )} \sqrt {2 \, \sin \left (x\right ) + 1} + 12 \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {3}{4}}}{3 \, {\left (2 \, \sin \left (x\right ) + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (48) = 96\).
Time = 37.51 (sec) , antiderivative size = 230, normalized size of antiderivative = 4.18 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=\frac {4 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \sin ^{2}{\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {2 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \sin {\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} + \frac {3 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \cos ^{2}{\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {2 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {24 \sin {\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {12}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} \]
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Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=\frac {1}{12} \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {3}{2}} - \frac {16 \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {1}{4}} - 3}{4 \, \sqrt {2 \, \sin \left (x\right ) + 1}} - \frac {1}{2} \, \sqrt {2 \, \sin \left (x\right ) + 1} \]
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=\frac {1}{12} \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {3}{2}} - \frac {16 \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {1}{4}} - 3}{4 \, \sqrt {2 \, \sin \left (x\right ) + 1}} - \frac {1}{2} \, \sqrt {2 \, \sin \left (x\right ) + 1} \]
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Timed out. \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx=-\int -\frac {\cos \left (x\right )\,\left (2\,{\left (2\,\sin \left (x\right )+1\right )}^{1/4}-{\cos \left (x\right )}^2\right )}{{\left (2\,\sin \left (x\right )+1\right )}^{3/2}} \,d x \]
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