Optimal. Leaf size=58 \[ \frac {1}{4} \sin (x) \left (-4 \sin ^2(x)-1\right )^{3/2}-\frac {3}{8} \sin (x) \sqrt {-4 \sin ^2(x)-1}+\frac {3}{16} \tan ^{-1}\left (\frac {2 \sin (x)}{\sqrt {-4 \sin ^2(x)-1}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4356, 195, 217, 203} \[ \frac {1}{4} \sin (x) \left (-4 \sin ^2(x)-1\right )^{3/2}-\frac {3}{8} \sin (x) \sqrt {-4 \sin ^2(x)-1}+\frac {3}{16} \tan ^{-1}\left (\frac {2 \sin (x)}{\sqrt {-4 \sin ^2(x)-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 4356
Rubi steps
\begin {align*} \int \cos (x) \left (-\cos ^2(x)-5 \sin ^2(x)\right )^{3/2} \, dx &=\operatorname {Subst}\left (\int \left (-1-4 x^2\right )^{3/2} \, dx,x,\sin (x)\right )\\ &=\frac {1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}-\frac {3}{4} \operatorname {Subst}\left (\int \sqrt {-1-4 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {3}{8} \sin (x) \sqrt {-1-4 \sin ^2(x)}+\frac {1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-4 x^2}} \, dx,x,\sin (x)\right )\\ &=-\frac {3}{8} \sin (x) \sqrt {-1-4 \sin ^2(x)}+\frac {1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {\sin (x)}{\sqrt {-1-4 \sin ^2(x)}}\right )\\ &=\frac {3}{16} \tan ^{-1}\left (\frac {2 \sin (x)}{\sqrt {-1-4 \sin ^2(x)}}\right )-\frac {3}{8} \sin (x) \sqrt {-1-4 \sin ^2(x)}+\frac {1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 61, normalized size = 1.05 \[ \frac {\sqrt {2 \cos (2 x)-3} \left (2 (2 \sin (3 x)-11 \sin (x)) \sqrt {3-2 \cos (2 x)}-3 \sinh ^{-1}(2 \sin (x))\right )}{16 \sqrt {4 \sin ^2(x)+1}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (x) \left (-\cos ^2(x)-5 \sin ^2(x)\right )^{3/2} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 1.31, size = 122, normalized size = 2.10 \[ \frac {1}{128} \, {\left (12 i \, e^{\left (4 i \, x\right )} \log \left (-\frac {1}{2} \, \sqrt {e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1} {\left (4 \, e^{\left (2 i \, x\right )} - 5\right )} + 2 \, e^{\left (4 i \, x\right )} - \frac {11}{2} \, e^{\left (2 i \, x\right )} + \frac {5}{2}\right ) - 12 i \, e^{\left (4 i \, x\right )} \log \left (\sqrt {e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} - 1\right ) + {\left (-16 i \, e^{\left (6 i \, x\right )} + 88 i \, e^{\left (4 i \, x\right )} - 88 i \, e^{\left (2 i \, x\right )} + 16 i\right )} \sqrt {e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1} - 145 i \, e^{\left (4 i \, x\right )}\right )} e^{\left (-4 i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.64, size = 41, normalized size = 0.71 \[ -\frac {1}{8} i \, {\left (8 \, \sin \relax (x)^{2} + 5\right )} \sqrt {4 \, \sin \relax (x)^{2} + 1} \sin \relax (x) + \frac {3}{16} i \, \log \left (\sqrt {4 \, \sin \relax (x)^{2} + 1} - 2 \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 82, normalized size = 1.41
method | result | size |
default | \(\frac {\sqrt {\left (4 \left (\cos ^{2}\relax (x )\right )-5\right ) \left (\sin ^{2}\relax (x )\right )}\, \left (-32 \sqrt {-4 \left (\sin ^{4}\relax (x )\right )-\left (\sin ^{2}\relax (x )\right )}\, \left (\sin ^{2}\relax (x )\right )-20 \sqrt {-4 \left (\sin ^{4}\relax (x )\right )-\left (\sin ^{2}\relax (x )\right )}+3 \arcsin \left (8 \left (\sin ^{2}\relax (x )\right )+1\right )\right )}{32 \sin \relax (x ) \sqrt {4 \left (\cos ^{2}\relax (x )\right )-5}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.21, size = 36, normalized size = 0.62 \[ \frac {1}{4} \, {\left (-4 \, \sin \relax (x)^{2} - 1\right )}^{\frac {3}{2}} \sin \relax (x) - \frac {3}{8} \, \sqrt {-4 \, \sin \relax (x)^{2} - 1} \sin \relax (x) - \frac {3}{16} i \, \operatorname {arsinh}\left (2 \, \sin \relax (x)\right ) \]
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 \[ \int \cos \relax (x)\,{\left (-{\cos \relax (x)}^2-5\,{\sin \relax (x)}^2\right )}^{3/2} \,d x \]
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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