Optimal. Leaf size=87 \[ -\frac {11 \cos (x)}{20 \cos ^{\frac {3}{2}}(2 x)}-\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {63 \cos (x)}{20 \sqrt {\cos (2 x)}}+\frac {3 \sin ^2(x) \cos (x)}{10 \cos ^{\frac {5}{2}}(2 x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.21, antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {4377, 12, 452, 288, 217, 206, 4366, 378, 191} \[ -\frac {2 \cos ^3(x)}{3 \cos ^{\frac {3}{2}}(2 x)}+\frac {13 \cos (x)}{5 \sqrt {\cos (2 x)}}+\frac {3 \sin ^4(x) \cos (x)}{5 \cos ^{\frac {5}{2}}(2 x)}-\frac {4 \sin ^2(x) \cos (x)}{5 \cos ^{\frac {3}{2}}(2 x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 206
Rule 217
Rule 288
Rule 378
Rule 452
Rule 4366
Rule 4377
Rubi steps
\begin {align*} \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx &=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 \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (-1+2 x^2\right )^{7/2}} \, dx,x,\cos (x)\right )\right )+\operatorname {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} \operatorname {Subst}\left (\int \frac {1-x^2}{\left (-1+2 x^2\right )^{5/2}} \, dx,x,\cos (x)\right )+4 \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x^2\right )^{3/2}} \, dx,x,\cos (x)\right )-2 \operatorname {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)}-\operatorname {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)}-\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right )\\ &=-\frac {\tanh ^{-1}\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*}
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Mathematica [A] time = 0.20, size = 62, normalized size = 0.71 \[ \frac {250 \cos (x)+45 \cos (3 x)+169 \cos (5 x)-120 \sqrt {2} \cos ^{\frac {5}{2}}(2 x) \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{240 \cos ^{\frac {5}{2}}(2 x)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^2(x) \left (3 \sin ^3(x)-\cos (x) \sin (4 x)\right )}{\cos ^{\frac {7}{2}}(2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.37, size = 163, normalized size = 1.87 \[ \frac {15 \, {\left (8 \, \sqrt {2} \cos \relax (x)^{6} - 12 \, \sqrt {2} \cos \relax (x)^{4} + 6 \, \sqrt {2} \cos \relax (x)^{2} - \sqrt {2}\right )} \log \left (2048 \, \cos \relax (x)^{8} - 2048 \, \cos \relax (x)^{6} + 640 \, \cos \relax (x)^{4} - 64 \, \cos \relax (x)^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \relax (x)^{7} - 96 \, \sqrt {2} \cos \relax (x)^{5} + 20 \, \sqrt {2} \cos \relax (x)^{3} - \sqrt {2} \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1} + 1\right ) + 16 \, {\left (169 \, \cos \relax (x)^{5} - 200 \, \cos \relax (x)^{3} + 60 \, \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1}}{240 \, {\left (8 \, \cos \relax (x)^{6} - 12 \, \cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 55, normalized size = 0.63 \[ \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {2} \cos \relax (x) + \sqrt {2 \, \cos \relax (x)^{2} - 1} \right |}\right ) + \frac {{\left ({\left (169 \, \cos \relax (x)^{2} - 200\right )} \cos \relax (x)^{2} + 60\right )} \cos \relax (x)}{15 \, {\left (2 \, \cos \relax (x)^{2} - 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 180, normalized size = 2.07
method | result | size |
default | \(-\frac {120 \ln \left (\cos \relax (x ) \sqrt {2}+\sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\right ) \sqrt {2}\, \left (\sin ^{6}\relax (x )\right )+338 \sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\, \cos \relax (x ) \left (\sin ^{4}\relax (x )\right )-180 \ln \left (\cos \relax (x ) \sqrt {2}+\sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\right ) \sqrt {2}\, \left (\sin ^{4}\relax (x )\right )-276 \sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\, \left (\sin ^{2}\relax (x )\right ) \cos \relax (x )+90 \ln \left (\cos \relax (x ) \sqrt {2}+\sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\right ) \sqrt {2}\, \left (\sin ^{2}\relax (x )\right )+58 \sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\, \cos \relax (x )-15 \ln \left (\cos \relax (x ) \sqrt {2}+\sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\right ) \sqrt {2}}{30 \left (8 \left (\sin ^{6}\relax (x )\right )-12 \left (\sin ^{4}\relax (x )\right )+6 \left (\sin ^{2}\relax (x )\right )-1\right )}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.61, size = 1359, normalized size = 15.62 \[ \text {result too large to display} \]
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 \[ \int \frac {{\sin \relax (x)}^2\,\left (3\,{\sin \relax (x)}^3-\sin \left (4\,x\right )\,\cos \relax (x)\right )}{{\cos \left (2\,x\right )}^{7/2}} \,d x \]
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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