Optimal. Leaf size=102 \[ \frac {1}{36} \cos ^{\frac {9}{4}}(2 x)-\frac {1}{5} \cos ^{\frac {5}{4}}(2 x)+\frac {7}{4} \sqrt [4]{\cos (2 x)}+\frac {\tan ^{-1}\left (\frac {1-\sqrt {\cos (2 x)}}{\sqrt {2} \sqrt [4]{\cos (2 x)}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\cos (2 x)}+1}{\sqrt {2} \sqrt [4]{\cos (2 x)}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.19, antiderivative size = 154, normalized size of antiderivative = 1.51, number of steps used = 14, number of rules used = 10, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4361, 446, 88, 63, 211, 1165, 628, 1162, 617, 204} \[ \frac {1}{36} \cos ^{\frac {9}{4}}(2 x)-\frac {1}{5} \cos ^{\frac {5}{4}}(2 x)+\frac {7}{4} \sqrt [4]{\cos (2 x)}+\frac {\log \left (\sqrt {\cos (2 x)}-\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {\cos (2 x)}+\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{\cos (2 x)}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 204
Rule 211
Rule 446
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 4361
Rubi steps
\begin {align*} \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx &=-\operatorname {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} \operatorname {Subst}\left (\int \frac {(1-x)^3}{x (-1+2 x)^{3/4}} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {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} \operatorname {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}-\operatorname {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} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+2 \cos ^2(x)}\right )+\frac {\operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}}\\ &=\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+2 \cos ^2(x)}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\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*}
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Mathematica [A] time = 0.13, size = 153, normalized size = 1.50 \[ \frac {1}{360} \left (-72 \cos ^{\frac {5}{4}}(2 x)+5 \cos (4 x) \sqrt [4]{\cos (2 x)}+635 \sqrt [4]{\cos (2 x)}+90 \sqrt {2} \log \left (\sqrt {\cos (2 x)}-\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )-90 \sqrt {2} \log \left (\sqrt {\cos (2 x)}+\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )+180 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{\cos (2 x)}\right )-180 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt [4]{\cos (2 x)}+1\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^6(x) \tan (x)}{\cos ^{\frac {3}{4}}(2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 120, normalized size = 1.18 \[ \frac {1}{36} \, \cos \left (2 \, x\right )^{\frac {9}{4}} - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \cos \left (2 \, x\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \cos \left (2 \, x\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \cos \left (2 \, x\right )^{\frac {1}{4}} + \sqrt {\cos \left (2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \cos \left (2 \, x\right )^{\frac {1}{4}} + \sqrt {\cos \left (2 \, x\right )} + 1\right ) - \frac {1}{5} \, \cos \left (2 \, x\right )^{\frac {5}{4}} + \frac {7}{4} \, \cos \left (2 \, x\right )^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {\left (\sin ^{6}\relax (x )\right ) \tan \relax (x )}{\cos \left (2 x \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \relax (x)^{6} \tan \relax (x)}{\cos \left (2 \, x\right )^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \relax (x)}^6\,\mathrm {tan}\relax (x)}{{\cos \left (2\,x\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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