Optimal. Leaf size=58 \[ \frac {3}{16} \tan ^{-1}\left (\frac {2 \sin (x)}{\sqrt {-1-4 \sin ^2(x)}}\right )-\frac {3}{8} \sin (x) \sqrt {-1-4 \sin ^2(x)}+\frac {1}{4} \sin (x) \left (-1-4 \sin ^2(x)\right )^{3/2} \]
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Rubi [A]
time = 0.04, 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 = {4441, 201, 223,
209} \begin {gather*} \frac {3}{16} \text {ArcTan}\left (\frac {2 \sin (x)}{\sqrt {-4 \sin ^2(x)-1}}\right )+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 4441
Rubi steps
\begin {align*} \int \cos (x) \left (-\cos ^2(x)-5 \sin ^2(x)\right )^{3/2} \, dx &=\text {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} \text {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} \text {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} \text {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 = 0.05, size = 61, normalized size = 1.05 \begin {gather*} \frac {\sqrt {-3+2 \cos (2 x)} \left (-3 \sinh ^{-1}(2 \sin (x))+2 \sqrt {3-2 \cos (2 x)} (-11 \sin (x)+2 \sin (3 x))\right )}{16 \sqrt {1+4 \sin ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 82, normalized size = 1.41
method | result | size |
default | \(\frac {\sqrt {\left (4 \left (\cos ^{2}\left (x \right )\right )-5\right ) \left (\sin ^{2}\left (x \right )\right )}\, \left (-32 \sqrt {-4 \left (\sin ^{4}\left (x \right )\right )-\left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right )-20 \sqrt {-4 \left (\sin ^{4}\left (x \right )\right )-\left (\sin ^{2}\left (x \right )\right )}+3 \arcsin \left (8 \left (\sin ^{2}\left (x \right )\right )+1\right )\right )}{32 \sin \left (x \right ) \sqrt {4 \left (\cos ^{2}\left (x \right )\right )-5}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 3.28, size = 36, normalized size = 0.62 \begin {gather*} \frac {1}{4} \, {\left (-4 \, \sin \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \sin \left (x\right ) - \frac {3}{8} \, \sqrt {-4 \, \sin \left (x\right )^{2} - 1} \sin \left (x\right ) - \frac {3}{16} i \, \operatorname {arsinh}\left (2 \, \sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.23, size = 123, normalized size = 2.12 \begin {gather*} \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 ) - 8 \, {\left (2 i \, e^{\left (6 i \, x\right )} - 11 i \, e^{\left (4 i \, x\right )} + 11 i \, e^{\left (2 i \, x\right )} - 2 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.14, size = 41, normalized size = 0.71 \begin {gather*} -\frac {1}{8} i \, {\left (8 \, \sin \left (x\right )^{2} + 5\right )} \sqrt {4 \, \sin \left (x\right )^{2} + 1} \sin \left (x\right ) + \frac {3}{16} i \, \log \left (\sqrt {4 \, \sin \left (x\right )^{2} + 1} - 2 \, \sin \left (x\right )\right ) \end {gather*}
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 \begin {gather*} \int \cos \left (x\right )\,{\left (-{\cos \left (x\right )}^2-5\,{\sin \left (x\right )}^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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