Optimal. Leaf size=108 \[ -\frac {\sqrt {\sin (2 x)} \cos (x) \sin ^{-1}(\cos (x)-\sin (x))}{\sqrt {\sin (x) \cos ^3(x)}}-\frac {\sin (2 x)}{\sqrt {\sin (x) \cos ^3(x)}}-\frac {\sqrt {\sin (2 x)} \cos (x) \tanh ^{-1}(\sin (x))}{\sqrt {\sin (x) \cos ^3(x)}}-\sqrt {2} \log \left (\sin (x)+\cos (x)-\sqrt {2} \sec (x) \sqrt {\sin (x) \cos ^3(x)}\right ) \]
________________________________________________________________________________________
Rubi [B] time = 1.54, 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 = {6719, 6725, 215, 329, 211, 1165, 628, 1162, 617, 204, 321} \[ -2 \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}-\frac {\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {2} \sqrt {\tan (x)}}-\sqrt {2} \cot (x) \sec ^2(x)^{3/2} \sqrt {\sin (x) \cos (x)} \sqrt {\sin (x) \cos ^3(x)} \sinh ^{-1}(\tan (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 215
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6719
Rule 6725
Rubi steps
\begin {align*} \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx &=\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [C] time = 0.28, size = 105, normalized size = 0.97 \[ \frac {-4 \sin (x) \cos ^3(x) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\cos ^2(x)\right )-3 \sqrt [4]{\sin ^2(x)} \cos (x) \left (2 \sin (x)+\sqrt {\sin (2 x)} \left (\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )\right )\right )}{3 \sqrt [4]{\sin ^2(x)} \sqrt {\sin (x) \cos ^3(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 20.64, size = 611, normalized size = 5.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\sqrt {\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt {\cos \relax (x)^{3} \sin \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.76, size = 247, normalized size = 2.29
method | result | size |
default | \(-\frac {2 \cos \relax (x ) \sin \relax (x )}{\sqrt {\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}}+\frac {2 \sqrt {2}\, \cos \relax (x ) \sqrt {\cos \relax (x ) \sin \relax (x )}\, \arctanh \left (\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )}{\sqrt {\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}}-\frac {\sqrt {2}\, \left (i \EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-2 \EllipticF \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )\right ) \cos \relax (x ) \sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \left (\sin ^{2}\relax (x )\right )}{\left (-1+\cos \relax (x )\right ) \sqrt {\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}}\) | \(247\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (2\,x\right )-\sqrt {\sin \left (2\,x\right )}}{\sqrt {{\cos \relax (x)}^3\,\sin \relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________