Optimal. Leaf size=108 \[ -\sqrt {2} \log \left (\cos (x)+\sin (x)-\sqrt {2} \sec (x) \sqrt {\cos ^3(x) \sin (x)}\right )-\frac {\sin ^{-1}(\cos (x)-\sin (x)) \cos (x) \sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}}-\frac {\tanh ^{-1}(\sin (x)) \cos (x) \sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}}-\frac {\sin (2 x)}{\sqrt {\cos ^3(x) \sin (x)}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(108)=216\).
time = 1.08, antiderivative size = 234, normalized size of antiderivative = 2.17, number of steps
used = 27, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6851, 6857,
221, 335, 217, 1179, 642, 1176, 631, 210, 327} \begin {gather*} -\frac {\sqrt {2} \sec ^2(x) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \sec ^2(x) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-2 \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}-\frac {\sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {2} \sqrt {\tan (x)}}-\sqrt {2} \cot (x) \sec ^2(x)^{3/2} \sqrt {\sin (x) \cos (x)} \sqrt {\sin (x) \cos ^3(x)} \sinh ^{-1}(\tan (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 221
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6851
Rule 6857
Rubi steps
\begin {align*} \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx &=\text {Subst}\left (\int \frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}} \left (1-x^2-\frac {x}{\sqrt {\frac {x}{2+2 x^2}}}\right )}{x} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1-x^2-\frac {x}{\sqrt {\frac {x}{2+2 x^2}}}}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}\\ &=\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {2} \sqrt {\frac {x}{1+x^2}}}{\sqrt {x}}+\frac {1}{\sqrt {x} \left (1+x^2\right )}-\frac {x^{3/2}}{1+x^2}\right ) \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}\\ &=\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}-\frac {\left (\sqrt {2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {x}{1+x^2}}}{\sqrt {x}} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\left (\sqrt {2} \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tan (x)\right )+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2} \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2} \sqrt {\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2 \sqrt {2} \sqrt {\tan (x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2 \sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2} \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}-\frac {\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.19, size = 105, normalized size = 0.97 \begin {gather*} \frac {-4 \cos ^3(x) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\cos ^2(x)\right ) \sin (x)-3 \cos (x) \sqrt [4]{\sin ^2(x)} \left (2 \sin (x)+\left (-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right ) \sqrt {\sin (2 x)}\right )}{3 \sqrt {\cos ^3(x) \sin (x)} \sqrt [4]{\sin ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.52, size = 247, normalized size = 2.29
method | result | size |
default | \(-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {2 \sqrt {2}\, \arctanh \left (\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right ) \cos \left (x \right ) \sqrt {\cos \left (x \right ) \sin \left (x \right )}}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}-\frac {\sqrt {2}\, \left (i \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-2 \EllipticF \left (\sqrt {-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )\right ) \cos \left (x \right ) \left (\sin ^{2}\left (x \right )\right ) \sqrt {-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}}\, \sqrt {\frac {-1+\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}}\, \sqrt {\frac {\cos \left (x \right )-1}{\sin \left (x \right )}}}{\left (\cos \left (x \right )-1\right ) \sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}\) | \(247\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 611 vs.
\(2 (92) = 184\).
time = 11.47, size = 611, normalized size = 5.66 \begin {gather*} -\frac {2 \, \sqrt {2} \arctan \left (-\frac {2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} \sin \left (x\right ) - 2 \, \cos \left (x\right )^{2} - \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} \sqrt {\frac {4 \, \cos \left (x\right )^{2} \sin \left (x\right ) + 2 \, \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \cos \left (x\right )}{\cos \left (x\right )}} - \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}}{2 \, {\left (\cos \left (x\right )^{4} + \cos \left (x\right )^{3} \sin \left (x\right ) - \cos \left (x\right )^{2}\right )}}\right ) \cos \left (x\right )^{2} + 2 \, \sqrt {2} \arctan \left (\frac {2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} \sin \left (x\right ) - 2 \, \cos \left (x\right )^{2} + \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} \sqrt {\frac {4 \, \cos \left (x\right )^{2} \sin \left (x\right ) - 2 \, \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \cos \left (x\right )}{\cos \left (x\right )}} + \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}}{2 \, {\left (\cos \left (x\right )^{4} + \cos \left (x\right )^{3} \sin \left (x\right ) - \cos \left (x\right )^{2}\right )}}\right ) \cos \left (x\right )^{2} - 2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + {\left (2 \, \cos \left (x\right )^{2} \sin \left (x\right ) - \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )}\right )} \sqrt {\frac {4 \, \cos \left (x\right )^{2} \sin \left (x\right ) + 2 \, \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \cos \left (x\right )}{\cos \left (x\right )}}}{2 \, \cos \left (x\right )^{2} \sin \left (x\right )}\right ) \cos \left (x\right )^{2} - 2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} - {\left (2 \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )}\right )} \sqrt {\frac {4 \, \cos \left (x\right )^{2} \sin \left (x\right ) - 2 \, \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \cos \left (x\right )}{\cos \left (x\right )}}}{2 \, \cos \left (x\right )^{2} \sin \left (x\right )}\right ) \cos \left (x\right )^{2} - \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {4 \, \cos \left (x\right )^{2} \sin \left (x\right ) + 2 \, \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \cos \left (x\right )}{\cos \left (x\right )}\right ) + \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {4 \, \cos \left (x\right )^{2} \sin \left (x\right ) - 2 \, \sqrt {2} \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \cos \left (x\right )}{\cos \left (x\right )}\right ) - \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {\cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + 4 \, \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\cos \left (x\right )^{2} - 2\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 8 \, \cos \left (x\right )^{2}}{\cos \left (x\right )^{6}}\right ) + 8 \, \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}}{4 \, \cos \left (x\right )^{2}} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {\cos \left (2\,x\right )-\sqrt {\sin \left (2\,x\right )}}{\sqrt {{\cos \left (x\right )}^3\,\sin \left (x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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