Integrand size = 28, antiderivative size = 87 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}}-\frac {11 \cos (x)}{20 \cos ^{\frac {3}{2}}(2 x)}-\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {63 \cos (x)}{20 \sqrt {\cos (2 x)}}+\frac {3 \cos (x) \sin ^2(x)}{10 \cos ^{\frac {5}{2}}(2 x)} \]
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Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {4462, 12, 463, 294, 223, 212, 4451, 386, 197} \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}}-\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {13 \cos (x)}{5 \sqrt {\cos (2 x)}}+\frac {3 \sin ^4(x) \cos (x)}{5 \cos ^{\frac {5}{2}}(2 x)}-\frac {4 \sin ^2(x) \cos (x)}{5 \cos ^{\frac {3}{2}}(2 x)} \]
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Rule 12
Rule 197
Rule 212
Rule 223
Rule 294
Rule 386
Rule 463
Rule 4451
Rule 4462
Rubi steps \begin{align*} \text {integral}& = 3 \int \frac {\sin ^5(x)}{\cos ^{\frac {7}{2}}(2 x)} \, dx-\int \frac {\cos (x) \sin ^2(x) \sin (4 x)}{\cos ^{\frac {7}{2}}(2 x)} \, dx \\ & = -\left (3 \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (-1+2 x^2\right )^{7/2}} \, dx,x,\cos (x)\right )\right )+\text {Subst}\left (\int \frac {4 x^2 \left (1-x^2\right )}{\left (-1+2 x^2\right )^{5/2}} \, dx,x,\cos (x)\right ) \\ & = \frac {3 \cos (x) \sin ^4(x)}{5 \cos ^{\frac {5}{2}}(2 x)}+\frac {12}{5} \text {Subst}\left (\int \frac {1-x^2}{\left (-1+2 x^2\right )^{5/2}} \, dx,x,\cos (x)\right )+4 \text {Subst}\left (\int \frac {x^2 \left (1-x^2\right )}{\left (-1+2 x^2\right )^{5/2}} \, dx,x,\cos (x)\right ) \\ & = -\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}-\frac {4 \cos (x) \sin ^2(x)}{5 \cos ^{\frac {3}{2}}(2 x)}+\frac {3 \cos (x) \sin ^4(x)}{5 \cos ^{\frac {5}{2}}(2 x)}-\frac {8}{5} \text {Subst}\left (\int \frac {1}{\left (-1+2 x^2\right )^{3/2}} \, dx,x,\cos (x)\right )-2 \text {Subst}\left (\int \frac {x^2}{\left (-1+2 x^2\right )^{3/2}} \, dx,x,\cos (x)\right ) \\ & = -\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {13 \cos (x)}{5 \sqrt {\cos (2 x)}}-\frac {4 \cos (x) \sin ^2(x)}{5 \cos ^{\frac {3}{2}}(2 x)}+\frac {3 \cos (x) \sin ^4(x)}{5 \cos ^{\frac {5}{2}}(2 x)}-\text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,\cos (x)\right ) \\ & = -\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {13 \cos (x)}{5 \sqrt {\cos (2 x)}}-\frac {4 \cos (x) \sin ^2(x)}{5 \cos ^{\frac {3}{2}}(2 x)}+\frac {3 \cos (x) \sin ^4(x)}{5 \cos ^{\frac {5}{2}}(2 x)}-\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}}-\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {13 \cos (x)}{5 \sqrt {\cos (2 x)}}-\frac {4 \cos (x) \sin ^2(x)}{5 \cos ^{\frac {3}{2}}(2 x)}+\frac {3 \cos (x) \sin ^4(x)}{5 \cos ^{\frac {5}{2}}(2 x)} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\frac {250 \cos (x)+45 \cos (3 x)+169 \cos (5 x)-120 \sqrt {2} \cos ^{\frac {5}{2}}(2 x) \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{240 \cos ^{\frac {5}{2}}(2 x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(65)=130\).
Time = 2.61 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {120 \sqrt {2}\, \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \left (\sin ^{6}\left (x \right )\right )+338 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right ) \left (\sin ^{4}\left (x \right )\right )-180 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{4}\left (x \right )\right )-276 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )+90 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{2}\left (x \right )\right )+58 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right )-15 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}}{30 \left (8 \left (\sin ^{6}\left (x \right )\right )-12 \left (\sin ^{4}\left (x \right )\right )+6 \left (\sin ^{2}\left (x \right )\right )-1\right )}\) | \(180\) |
parts | \(-\frac {\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right ) \left (43 \left (\sin ^{4}\left (x \right )\right )-36 \left (\sin ^{2}\left (x \right )\right )+8\right )}{5 \left (8 \left (\sin ^{6}\left (x \right )\right )-12 \left (\sin ^{4}\left (x \right )\right )+6 \left (\sin ^{2}\left (x \right )\right )-1\right )}-\frac {12 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{4}\left (x \right )\right )+8 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )-12 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}\, \left (\sin ^{2}\left (x \right )\right )-2 \sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right )+3 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\right ) \sqrt {2}}{6 \left (4 \left (\sin ^{4}\left (x \right )\right )-4 \left (\sin ^{2}\left (x \right )\right )+1\right )}\) | \(180\) |
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (65) = 130\).
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.87 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\frac {15 \, {\left (8 \, \sqrt {2} \cos \left (x\right )^{6} - 12 \, \sqrt {2} \cos \left (x\right )^{4} + 6 \, \sqrt {2} \cos \left (x\right )^{2} - \sqrt {2}\right )} \log \left (2048 \, \cos \left (x\right )^{8} - 2048 \, \cos \left (x\right )^{6} + 640 \, \cos \left (x\right )^{4} - 64 \, \cos \left (x\right )^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \left (x\right )^{7} - 96 \, \sqrt {2} \cos \left (x\right )^{5} + 20 \, \sqrt {2} \cos \left (x\right )^{3} - \sqrt {2} \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1} + 1\right ) + 16 \, {\left (169 \, \cos \left (x\right )^{5} - 200 \, \cos \left (x\right )^{3} + 60 \, \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1}}{240 \, {\left (8 \, \cos \left (x\right )^{6} - 12 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} - 1\right )}} \]
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Timed out. \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1359 vs. \(2 (65) = 130\).
Time = 0.41 (sec) , antiderivative size = 1359, normalized size of antiderivative = 15.62 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {2} \cos \left (x\right ) + \sqrt {2 \, \cos \left (x\right )^{2} - 1} \right |}\right ) + \frac {{\left ({\left (169 \, \cos \left (x\right )^{2} - 200\right )} \cos \left (x\right )^{2} + 60\right )} \cos \left (x\right )}{15 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx=\int \frac {{\sin \left (x\right )}^2\,\left (3\,{\sin \left (x\right )}^3-\sin \left (4\,x\right )\,\cos \left (x\right )\right )}{{\cos \left (2\,x\right )}^{7/2}} \,d x \]
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