Optimal. Leaf size=49 \[ 2 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )-\frac {5}{2} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\frac {1}{2} \sqrt {\cos (2 x)} \tan (x) \sec (x) \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {4364, 413, 523, 216, 377, 203} \[ 2 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )-\frac {5}{2} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\frac {1}{2} \sqrt {\cos (2 x)} \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 216
Rule 377
Rule 413
Rule 523
Rule 4364
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-2 x^2\right )^{3/2}}{\left (1-x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-3+8 x^2}{\sqrt {1-2 x^2} \left (1-x^2\right )} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2} \left (1-x^2\right )} \, dx,x,\sin (x)\right )+4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2}} \, dx,x,\sin (x)\right )\\ &=2 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )\\ &=2 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )-\frac {5}{2} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 49, normalized size = 1.00 \[ \frac {1}{2} \left (4 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )-5 \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\sqrt {\cos (2 x)} \tan (x) \sec (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.97, size = 118, normalized size = 2.41 \[ -\frac {2 \, \sqrt {2} \arctan \left (\frac {{\left (32 \, \sqrt {2} \cos \relax (x)^{4} - 48 \, \sqrt {2} \cos \relax (x)^{2} + 17 \, \sqrt {2}\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1}}{8 \, {\left (8 \, \cos \relax (x)^{4} - 10 \, \cos \relax (x)^{2} + 3\right )} \sin \relax (x)}\right ) \cos \relax (x)^{2} - 5 \, \arctan \left (\frac {3 \, \cos \relax (x)^{2} - 2}{2 \, \sqrt {2 \, \cos \relax (x)^{2} - 1} \sin \relax (x)}\right ) \cos \relax (x)^{2} + 2 \, \sqrt {2 \, \cos \relax (x)^{2} - 1} \sin \relax (x)}{4 \, \cos \relax (x)^{2}} \]
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 \[ \int \frac {\cos \left (2 \, x\right )^{\frac {3}{2}}}{\cos \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 100, normalized size = 2.04
method | result | size |
default | \(-\frac {\sqrt {\left (2 \left (\cos ^{2}\relax (x )\right )-1\right ) \left (\sin ^{2}\relax (x )\right )}\, \left (4 \sqrt {2}\, \arcsin \left (4 \left (\cos ^{2}\relax (x )\right )-3\right ) \left (\cos ^{2}\relax (x )\right )-5 \arctan \left (\frac {3 \left (\cos ^{2}\relax (x )\right )-2}{2 \sqrt {-2 \left (\sin ^{4}\relax (x )\right )+\sin ^{2}\relax (x )}}\right ) \left (\cos ^{2}\relax (x )\right )+2 \sqrt {-2 \left (\sin ^{4}\relax (x )\right )+\sin ^{2}\relax (x )}\right )}{4 \cos \relax (x )^{2} \sin \relax (x ) \sqrt {2 \left (\cos ^{2}\relax (x )\right )-1}}\) | \(100\) |
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 \[ \int \frac {\cos \left (2 \, x\right )^{\frac {3}{2}}}{\cos \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\cos \left (2\,x\right )}^{3/2}}{{\cos \relax (x)}^3} \,d x \]
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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