3.416 \(\int \frac{\cos (2 x)-\sqrt{\sin (2 x)}}{\sqrt{\cos ^3(x) \sin (x)}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\sqrt{\sin (2 x)} \cos (x) \sin ^{-1}(\cos (x)-\sin (x))}{\sqrt{\sin (x) \cos ^3(x)}}-\frac{\sin (2 x)}{\sqrt{\sin (x) \cos ^3(x)}}-\frac{\sqrt{\sin (2 x)} \cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{\sin (x) \cos ^3(x)}}-\sqrt{2} \log \left (\sin (x)+\cos (x)-\sqrt{2} \sec (x) \sqrt{\sin (x) \cos ^3(x)}\right ) \]

[Out]

-(Sqrt[2]*Log[Cos[x] + Sin[x] - Sqrt[2]*Sec[x]*Sqrt[Cos[x]^3*Sin[x]]]) - (ArcSin
[Cos[x] - Sin[x]]*Cos[x]*Sqrt[Sin[2*x]])/Sqrt[Cos[x]^3*Sin[x]] - (ArcTanh[Sin[x]
]*Cos[x]*Sqrt[Sin[2*x]])/Sqrt[Cos[x]^3*Sin[x]] - Sin[2*x]/Sqrt[Cos[x]^3*Sin[x]]

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Rubi [B]  time = 2.76881, 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 \[ -2 \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}-\frac{\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}+\frac{\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (x)}+1\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{\sec ^2(x) \log \left (\tan (x)-\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{2} \sqrt{\tan (x)}}+\frac{\sec ^2(x) \log \left (\tan (x)+\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{2} \sqrt{\tan (x)}}-\sqrt{2} \cot (x) \sec ^2(x)^{3/2} \sqrt{\sin (x) \cos (x)} \sqrt{\sin (x) \cos ^3(x)} \sinh ^{-1}(\tan (x)) \]

Antiderivative was successfully verified.

[In]  Int[(Cos[2*x] - Sqrt[Sin[2*x]])/Sqrt[Cos[x]^3*Sin[x]],x]

[Out]

-2*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]] - Sqrt[2]*ArcSinh[Tan[x]]*Cot[x]*(Sec[x]^2)^(3
/2)*Sqrt[Cos[x]*Sin[x]]*Sqrt[Cos[x]^3*Sin[x]] - (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt
[Tan[x]]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/Sqrt[Tan[x]] + (Sqrt[2]*ArcTan[1 + Sqr
t[2]*Sqrt[Tan[x]]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/Sqrt[Tan[x]] - (Log[1 - Sqrt[
2]*Sqrt[Tan[x]] + Tan[x]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/(Sqrt[2]*Sqrt[Tan[x]])
+ (Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/(Sqrt
[2]*Sqrt[Tan[x]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((cos(2*x)-sin(2*x)**(1/2))/(cos(x)**3*sin(x))**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.363698, 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]  Integrate[(Cos[2*x] - Sqrt[Sin[2*x]])/Sqrt[Cos[x]^3*Sin[x]],x]

[Out]

(-4*Cos[x]^3*Hypergeometric2F1[3/4, 3/4, 7/4, Cos[x]^2]*Sin[x] - 3*Cos[x]*(Sin[x
]^2)^(1/4)*(2*Sin[x] + (-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sq
rt[Sin[2*x]]))/(3*Sqrt[Cos[x]^3*Sin[x]]*(Sin[x]^2)^(1/4))

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Maple [C]  time = 0.488, size = 244, normalized size = 2.3 \[ -2\,{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}}+2\,{\frac{\sqrt{2}\sqrt{\cos \left ( x \right ) \sin \left ( x \right ) }\cos \left ( x \right ) }{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}{\it Artanh} \left ({\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) }+{\frac{\cos \left ( x \right ) \sqrt{2} \left ( \sin \left ( x \right ) \right ) ^{2}}{\cos \left ( x \right ) -1} \left ( -i{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +i{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +2\,{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) \right ) \sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}\sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((cos(2*x)-sin(2*x)^(1/2))/(cos(x)^3*sin(x))^(1/2),x)

[Out]

-2*sin(x)*cos(x)/(cos(x)^3*sin(x))^(1/2)+2*2^(1/2)*(cos(x)*sin(x))^(1/2)*cos(x)*
arctanh((cos(x)-1)/sin(x))/(cos(x)^3*sin(x))^(1/2)+2^(1/2)*(-I*EllipticPi(((1-co
s(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+I*EllipticPi(((1-cos(x)+sin(x)
)/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+2*EllipticF(((1-cos(x)+sin(x))/sin(x))^(1
/2),1/2*2^(1/2))-EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/
2))-EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2)))*cos(x)*(
(sin(x)-1+cos(x))/sin(x))^(1/2)*((cos(x)-1)/sin(x))^(1/2)*sin(x)^2*((1-cos(x)+si
n(x))/sin(x))^(1/2)/(cos(x)-1)/(cos(x)^3*sin(x))^(1/2)

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Maxima [A]  time = 1.75785, size = 258, normalized size = 2.39 \[ \frac{3}{5} \, \tan \left (x\right )^{\frac{5}{2}} + \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) + \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, \sqrt{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \frac{{\left (\tan \left (x\right )^{3} + \tan \left (x\right )\right )} \tan \left (x\right )^{\frac{3}{2}}}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} - 4 \, \sqrt{\tan \left (x\right )} - \frac{{\left (\tan \left (x\right )^{3} - 15 \, \tan \left (x\right )\right )} \tan \left (x\right )^{\frac{3}{2}} - \frac{4 \,{\left (\tan \left (x\right )^{3} + 5 \, \tan \left (x\right )\right )}}{\sqrt{\tan \left (x\right )}}}{10 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(sin(2*x)) - cos(2*x))/sqrt(cos(x)^3*sin(x)),x, algorithm="maxima")

[Out]

3/5*tan(x)^(5/2) + sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(x)))) + sqrt
(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(x)))) - 1/2*sqrt(2)*log(cos(x)^2 +
sin(x)^2 + 2*sin(x) + 1) + 1/2*sqrt(2)*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)
+ 1/2*sqrt(2)*log(sqrt(2)*sqrt(tan(x)) + tan(x) + 1) - 1/2*sqrt(2)*log(-sqrt(2)*
sqrt(tan(x)) + tan(x) + 1) - 1/2*(tan(x)^3 + tan(x))*tan(x)^(3/2)/(tan(x)^2 + 1)
- 4*sqrt(tan(x)) - 1/10*((tan(x)^3 - 15*tan(x))*tan(x)^(3/2) - 4*(tan(x)^3 + 5*
tan(x))/sqrt(tan(x)))/(tan(x)^2 + 1)

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Fricas [A]  time = 0.531339, size = 761, normalized size = 7.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(sin(2*x)) - cos(2*x))/sqrt(cos(x)^3*sin(x)),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*(2*arctan(sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) - sin(x))/(sqrt(2)*s
qrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) - sqrt((4*cos(x)^2*sin(x) + 2*sqrt(2)*sqr
t(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x))*cos(x) + cos(x)))*cos(x)^
2 + 2*arctan(-sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) - sin(x))/(sqrt(2)*sqrt(cos(
x)^3*sin(x))*(cos(x) + sin(x)) + sqrt((4*cos(x)^2*sin(x) - 2*sqrt(2)*sqrt(cos(x)
^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x))*cos(x) - cos(x)))*cos(x)^2 + 2*ar
ctan((2*cos(x)^2*sin(x) + sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)))/(2*co
s(x)^3 + sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) - sin(x)) + sqrt((4*cos(x)^2*sin(
x) + 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x))*cos(x)
- cos(x)))*cos(x)^2 + 2*arctan(-(2*cos(x)^2*sin(x) - sqrt(2)*sqrt(cos(x)^3*sin(x
))*(cos(x) + sin(x)))/(2*cos(x)^3 - sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) - sin(
x)) + sqrt((4*cos(x)^2*sin(x) - 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)
) + cos(x))/cos(x))*cos(x) - cos(x)))*cos(x)^2 + cos(x)^2*log(2*(4*cos(x)^2*sin(
x) + 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x)) - cos(x
)^2*log(2*(4*cos(x)^2*sin(x) - 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x))
+ cos(x))/cos(x)) + cos(x)^2*log((cos(x)^6 - 8*cos(x)^4 + 4*sqrt(cos(x)^3*sin(x
))*(cos(x)^2 - 2)*sqrt(cos(x)*sin(x)) + 8*cos(x)^2)/cos(x)^6) - 4*sqrt(2)*sqrt(c
os(x)^3*sin(x)))/cos(x)^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(2*x)-sin(2*x)**(1/2))/(cos(x)**3*sin(x))**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt{\cos \left (x\right )^{3} \sin \left (x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(sin(2*x)) - cos(2*x))/sqrt(cos(x)^3*sin(x)),x, algorithm="giac")

[Out]

integrate(-(sqrt(sin(2*x)) - cos(2*x))/sqrt(cos(x)^3*sin(x)), x)