Optimal. Leaf size=68 \[ -\frac {9 \cos (x)}{16 \sqrt {\sin (2 x)}}+\frac {\cos (x) \cot ^2(x)}{20 \sqrt {\sin (2 x)}}-\frac {5 \cos (x) \cot (x)}{24 \sqrt {\sin (2 x)}}+\frac {33}{32} \tanh ^{-1}\left (\frac {1}{2} \sqrt {\sin (2 x)} \sec (x)\right ) \]
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Rubi [A] time = 0.86, antiderivative size = 95, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {4390, 1619, 63, 207} \[ \frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \sin (x) \cos ^4(x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {9 \sin ^2(x) \cos ^3(x)}{4 \sin ^{\frac {5}{2}}(2 x)}+\frac {33 \sin ^5(x) \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right )}{4 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 1619
Rule 4390
Rubi steps
\begin {align*} \int \frac {\cos ^3(x) (\cos (2 x)-3 \tan (x))}{\left (\sin ^2(x)-\sin (2 x)\right ) \sin ^{\frac {5}{2}}(2 x)} \, dx &=\frac {\sin ^5(x) \int \frac {\csc ^2(x) (\cos (2 x)-3 \tan (x))}{\left (\sin ^2(x)-\sin (2 x)\right ) \sqrt {\tan (x)}} \, dx}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\sin ^5(x) \operatorname {Subst}\left (\int \frac {-1+3 x+x^2+3 x^3}{(2-x) x^{7/2}} \, dx,x,\tan (x)\right )}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\sin ^5(x) \operatorname {Subst}\left (\int \left (-\frac {1}{2 x^{7/2}}+\frac {5}{4 x^{5/2}}+\frac {9}{8 x^{3/2}}-\frac {33}{8 (-2+x) \sqrt {x}}\right ) \, dx,x,\tan (x)\right )}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \cos ^4(x) \sin (x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {9 \cos ^3(x) \sin ^2(x)}{4 \sin ^{\frac {5}{2}}(2 x)}-\frac {\left (33 \sin ^5(x)\right ) \operatorname {Subst}\left (\int \frac {1}{(-2+x) \sqrt {x}} \, dx,x,\tan (x)\right )}{8 \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \cos ^4(x) \sin (x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {9 \cos ^3(x) \sin ^2(x)}{4 \sin ^{\frac {5}{2}}(2 x)}-\frac {\left (33 \sin ^5(x)\right ) \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{4 \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \cos ^4(x) \sin (x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {9 \cos ^3(x) \sin ^2(x)}{4 \sin ^{\frac {5}{2}}(2 x)}+\frac {33 \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right ) \sin ^5(x)}{4 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ \end {align*}
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Mathematica [C] time = 6.38, size = 150, normalized size = 2.21 \[ \frac {\sqrt {\sin (2 x)} \cos (x) (\cos (2 x)-3 \tan (x)) \left (\frac {1}{15} \csc (x) \left (-50 \cot (x)+12 \csc ^2(x)-147\right )-33 \sqrt {\frac {\cos (x)}{2 \cos (x)-2}} \sqrt {\tan \left (\frac {x}{2}\right )} \sec (x) \left (\operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right ),-1\right )-\Pi \left (-\frac {2}{-1+\sqrt {5}};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (-1+\sqrt {5}\right );\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )\right )\right )}{16 (-6 \sin (x)+\cos (x)+\cos (3 x))} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^3(x) (\cos (2 x)-3 \tan (x))}{\left (\sin ^2(x)-\sin (2 x)\right ) \sin ^{\frac {5}{2}}(2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.05, size = 136, normalized size = 2.00 \[ -\frac {495 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (4 \, \cos \relax (x) + 3 \, \sin \relax (x)\right )} + \frac {1}{2} \, \cos \relax (x)^{2} + \frac {7}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2}\right ) \sin \relax (x) - 495 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x)^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} \sin \relax (x) - \frac {1}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2}\right ) \sin \relax (x) + 4 \, \sqrt {2} {\left (147 \, \cos \relax (x)^{2} - 50 \, \cos \relax (x) \sin \relax (x) - 135\right )} \sqrt {\cos \relax (x) \sin \relax (x)} + 388 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}{1920 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\cos \left (2 \, x\right ) - 3 \, \tan \relax (x)\right )} \cos \relax (x)^{3}}{{\left (\sin \relax (x)^{2} - \sin \left (2 \, x\right )\right )} \sin \left (2 \, x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.61, size = 761, normalized size = 11.19
method | result | size |
default | \(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (932 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-3024 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+24 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+3 \sqrt {2}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+2 \textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (34 \underline {\hspace {1.25 ex}}\alpha ^{3}+13 \underline {\hspace {1.25 ex}}\alpha ^{2}+34 \underline {\hspace {1.25 ex}}\alpha -21\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha -3\right ) \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, -\frac {1}{4} \underline {\hspace {1.25 ex}}\alpha ^{3}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha +\frac {3}{4}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+200 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-24 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}-1920 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}-552 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-24 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+552 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-200 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \tan \left (\frac {x}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}+24 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\right )}{3840 \tan \left (\frac {x}{2}\right )^{3} \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\) | \(761\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {{\cos \relax (x)}^3\,\left (\cos \left (2\,x\right )-3\,\mathrm {tan}\relax (x)\right )}{{\sin \left (2\,x\right )}^{5/2}\,\left (\sin \left (2\,x\right )-{\sin \relax (x)}^2\right )} \,d x \]
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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