Integrand size = 15, antiderivative size = 73 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=-\frac {5 \text {arcsinh}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}-\frac {5}{16} \cos (x) \sqrt {1+2 \cos ^2(x)}-\frac {5}{24} \cos (x) \left (1+2 \cos ^2(x)\right )^{3/2}-\frac {1}{6} \cos (x) \left (1+2 \cos ^2(x)\right )^{5/2} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 201, 221} \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=-\frac {5 \text {arcsinh}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}-\frac {1}{6} \cos (x) (\cos (2 x)+2)^{5/2}-\frac {5}{24} \cos (x) (\cos (2 x)+2)^{3/2}-\frac {5}{16} \cos (x) \sqrt {\cos (2 x)+2} \]
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Rule 201
Rule 221
Rule 3269
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (1+2 x^2\right )^{5/2} \, dx,x,\cos (x)\right ) \\ & = -\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}-\frac {5}{6} \text {Subst}\left (\int \left (1+2 x^2\right )^{3/2} \, dx,x,\cos (x)\right ) \\ & = -\frac {5}{24} \cos (x) (2+\cos (2 x))^{3/2}-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}-\frac {5}{8} \text {Subst}\left (\int \sqrt {1+2 x^2} \, dx,x,\cos (x)\right ) \\ & = -\frac {5}{16} \cos (x) \sqrt {2+\cos (2 x)}-\frac {5}{24} \cos (x) (2+\cos (2 x))^{3/2}-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}-\frac {5}{16} \text {Subst}\left (\int \frac {1}{\sqrt {1+2 x^2}} \, dx,x,\cos (x)\right ) \\ & = -\frac {5 \text {arcsinh}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}-\frac {5}{16} \cos (x) \sqrt {2+\cos (2 x)}-\frac {5}{24} \cos (x) (2+\cos (2 x))^{3/2}-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=\frac {1}{96} \left (-2 \sqrt {2+\cos (2 x)} (92 \cos (x)+23 \cos (3 x)+2 \cos (5 x))-15 \sqrt {2} \log \left (\sqrt {2} \cos (x)+\sqrt {2+\cos (2 x)}\right )\right ) \]
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Time = 1.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {5 \cos \left (x \right ) {\left (1+2 \left (\cos ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}{24}-\frac {\cos \left (x \right ) {\left (1+2 \left (\cos ^{2}\left (x \right )\right )\right )}^{\frac {5}{2}}}{6}-\frac {5 \,\operatorname {arcsinh}\left (\cos \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{32}-\frac {5 \cos \left (x \right ) \sqrt {1+2 \left (\cos ^{2}\left (x \right )\right )}}{16}\) | \(56\) |
default | \(-\frac {5 \cos \left (x \right ) {\left (1+2 \left (\cos ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}{24}-\frac {\cos \left (x \right ) {\left (1+2 \left (\cos ^{2}\left (x \right )\right )\right )}^{\frac {5}{2}}}{6}-\frac {5 \,\operatorname {arcsinh}\left (\cos \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{32}-\frac {5 \cos \left (x \right ) \sqrt {1+2 \left (\cos ^{2}\left (x \right )\right )}}{16}\) | \(56\) |
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Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.48 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=-\frac {1}{48} \, {\left (32 \, \cos \left (x\right )^{5} + 52 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} + 1} + \frac {5}{256} \, \sqrt {2} \log \left (2048 \, \cos \left (x\right )^{8} + 2048 \, \cos \left (x\right )^{6} + 640 \, \cos \left (x\right )^{4} + 64 \, \cos \left (x\right )^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \left (x\right )^{7} + 96 \, \sqrt {2} \cos \left (x\right )^{5} + 20 \, \sqrt {2} \cos \left (x\right )^{3} + \sqrt {2} \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} + 1} + 1\right ) \]
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Timed out. \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=-\frac {1}{6} \, {\left (2 \, \cos \left (x\right )^{2} + 1\right )}^{\frac {5}{2}} \cos \left (x\right ) - \frac {5}{24} \, {\left (2 \, \cos \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \cos \left (x\right ) - \frac {5}{32} \, \sqrt {2} \operatorname {arsinh}\left (\sqrt {2} \cos \left (x\right )\right ) - \frac {5}{16} \, \sqrt {2 \, \cos \left (x\right )^{2} + 1} \cos \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=-\frac {1}{48} \, {\left (4 \, {\left (8 \, \cos \left (x\right )^{2} + 13\right )} \cos \left (x\right )^{2} + 33\right )} \sqrt {2 \, \cos \left (x\right )^{2} + 1} \cos \left (x\right ) + \frac {5}{32} \, \sqrt {2} \log \left (-\sqrt {2} \cos \left (x\right ) + \sqrt {2 \, \cos \left (x\right )^{2} + 1}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.59 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx=-\frac {5\,\sqrt {2}\,\mathrm {asinh}\left (\sqrt {2}\,\cos \left (x\right )\right )}{32}-\frac {\sqrt {2}\,\sqrt {{\cos \left (x\right )}^2+\frac {1}{2}}\,\left (\frac {4\,{\cos \left (x\right )}^5}{3}+\frac {13\,{\cos \left (x\right )}^3}{6}+\frac {11\,\cos \left (x\right )}{8}\right )}{2} \]
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