Integrand size = 15, antiderivative size = 102 \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\frac {\arctan \left (\frac {1-\sqrt {\cos (2 x)}}{\sqrt {2} \sqrt [4]{\cos (2 x)}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {1+\sqrt {\cos (2 x)}}{\sqrt {2} \sqrt [4]{\cos (2 x)}}\right )}{\sqrt {2}}+\frac {7}{4} \sqrt [4]{\cos (2 x)}-\frac {1}{5} \cos ^{\frac {5}{4}}(2 x)+\frac {1}{36} \cos ^{\frac {9}{4}}(2 x) \]
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Time = 0.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.51, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4446, 457, 90, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{\cos (2 x)}\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )}{\sqrt {2}}+\frac {1}{36} \cos ^{\frac {9}{4}}(2 x)-\frac {1}{5} \cos ^{\frac {5}{4}}(2 x)+\frac {7}{4} \sqrt [4]{\cos (2 x)}+\frac {\log \left (\sqrt {\cos (2 x)}-\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {\cos (2 x)}+\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )}{2 \sqrt {2}} \]
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Rule 65
Rule 90
Rule 210
Rule 217
Rule 457
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 4446
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x \left (-1+2 x^2\right )^{3/4}} \, dx,x,\cos (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {(1-x)^3}{x (-1+2 x)^{3/4}} \, dx,x,\cos ^2(x)\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \left (-\frac {7}{4 (-1+2 x)^{3/4}}+\frac {1}{x (-1+2 x)^{3/4}}+\sqrt [4]{-1+2 x}-\frac {1}{4} (-1+2 x)^{5/4}\right ) \, dx,x,\cos ^2(x)\right )\right ) \\ & = \frac {7}{4} \sqrt [4]{-1+2 \cos ^2(x)}-\frac {1}{5} \left (-1+2 \cos ^2(x)\right )^{5/4}+\frac {1}{36} \left (-1+2 \cos ^2(x)\right )^{9/4}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (-1+2 x)^{3/4}} \, dx,x,\cos ^2(x)\right ) \\ & = \frac {7}{4} \sqrt [4]{-1+2 \cos ^2(x)}-\frac {1}{5} \left (-1+2 \cos ^2(x)\right )^{5/4}+\frac {1}{36} \left (-1+2 \cos ^2(x)\right )^{9/4}-\text {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {x^4}{2}} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right ) \\ & = \frac {7}{4} \sqrt [4]{-1+2 \cos ^2(x)}-\frac {1}{5} \left (-1+2 \cos ^2(x)\right )^{5/4}+\frac {1}{36} \left (-1+2 \cos ^2(x)\right )^{9/4}-\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{\frac {1}{2}+\frac {x^4}{2}} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{\frac {1}{2}+\frac {x^4}{2}} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right ) \\ & = \frac {7}{4} \sqrt [4]{-1+2 \cos ^2(x)}-\frac {1}{5} \left (-1+2 \cos ^2(x)\right )^{5/4}+\frac {1}{36} \left (-1+2 \cos ^2(x)\right )^{9/4}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )}{2 \sqrt {2}} \\ & = \frac {7}{4} \sqrt [4]{-1+2 \cos ^2(x)}-\frac {1}{5} \left (-1+2 \cos ^2(x)\right )^{5/4}+\frac {1}{36} \left (-1+2 \cos ^2(x)\right )^{9/4}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}+\sqrt {-1+2 \cos ^2(x)}\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}+\sqrt {-1+2 \cos ^2(x)}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}} \\ & = \frac {\arctan \left (1-\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}}+\frac {7}{4} \sqrt [4]{-1+2 \cos ^2(x)}-\frac {1}{5} \left (-1+2 \cos ^2(x)\right )^{5/4}+\frac {1}{36} \left (-1+2 \cos ^2(x)\right )^{9/4}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}+\sqrt {-1+2 \cos ^2(x)}\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}+\sqrt {-1+2 \cos ^2(x)}\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.50 \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\frac {1}{360} \left (180 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt [4]{\cos (2 x)}\right )-180 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt [4]{\cos (2 x)}\right )+635 \sqrt [4]{\cos (2 x)}-72 \cos ^{\frac {5}{4}}(2 x)+5 \sqrt [4]{\cos (2 x)} \cos (4 x)+90 \sqrt {2} \log \left (1-\sqrt {2} \sqrt [4]{\cos (2 x)}+\sqrt {\cos (2 x)}\right )-90 \sqrt {2} \log \left (1+\sqrt {2} \sqrt [4]{\cos (2 x)}+\sqrt {\cos (2 x)}\right )\right ) \]
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\[\int \frac {\left (\sin ^{6}\left (x \right )\right ) \tan \left (x \right )}{\cos \left (2 x \right )^{\frac {3}{4}}}d x\]
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Timed out. \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\int { \frac {\sin \left (x\right )^{6} \tan \left (x\right )}{\cos \left (2 \, x\right )^{\frac {3}{4}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.18 \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\frac {1}{36} \, \cos \left (2 \, x\right )^{\frac {9}{4}} - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \cos \left (2 \, x\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \cos \left (2 \, x\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \cos \left (2 \, x\right )^{\frac {1}{4}} + \sqrt {\cos \left (2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \cos \left (2 \, x\right )^{\frac {1}{4}} + \sqrt {\cos \left (2 \, x\right )} + 1\right ) - \frac {1}{5} \, \cos \left (2 \, x\right )^{\frac {5}{4}} + \frac {7}{4} \, \cos \left (2 \, x\right )^{\frac {1}{4}} \]
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Timed out. \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx=\int \frac {{\sin \left (x\right )}^6\,\mathrm {tan}\left (x\right )}{{\cos \left (2\,x\right )}^{3/4}} \,d x \]
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