3.430 \(\int \cos ^{\frac{3}{2}}(2 x) \sin (x) \, dx\)

Optimal. Leaf size=55 \[ -\frac{1}{4} \cos (x) \cos ^{\frac{3}{2}}(2 x)+\frac{3}{8} \cos (x) \sqrt{\cos (2 x)}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)}}\right )}{8 \sqrt{2}} \]

[Out]

(-3*ArcTanh[(Sqrt[2]*Cos[x])/Sqrt[Cos[2*x]]])/(8*Sqrt[2]) + (3*Cos[x]*Sqrt[Cos[2
*x]])/8 - (Cos[x]*Cos[2*x]^(3/2))/4

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Rubi [A]  time = 0.0574155, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{1}{4} \cos (x) \cos ^{\frac{3}{2}}(2 x)+\frac{3}{8} \cos (x) \sqrt{\cos (2 x)}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)}}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Cos[2*x]^(3/2)*Sin[x],x]

[Out]

(-3*ArcTanh[(Sqrt[2]*Cos[x])/Sqrt[Cos[2*x]]])/(8*Sqrt[2]) + (3*Cos[x]*Sqrt[Cos[2
*x]])/8 - (Cos[x]*Cos[2*x]^(3/2))/4

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Rubi in Sympy [A]  time = 112.273, size = 65, normalized size = 1.18 \[ - \frac{\left (2 \cos ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}} \cos{\left (x \right )}}{4} + \frac{3 \sqrt{2 \cos ^{2}{\left (x \right )} - 1} \cos{\left (x \right )}}{8} - \frac{3 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \cos{\left (x \right )}}{\sqrt{2 \cos ^{2}{\left (x \right )} - 1}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*cos(x)**2 - 1)**(3/2)*cos(x)/4 + 3*sqrt(2*cos(x)**2 - 1)*cos(x)/8 - 3*sqrt(2
)*atanh(sqrt(2)*cos(x)/sqrt(2*cos(x)**2 - 1))/16

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Mathematica [A]  time = 0.121083, size = 49, normalized size = 0.89 \[ -\frac{1}{8} \sqrt{\cos (2 x)} (\cos (3 x)-2 \cos (x))-\frac{3 \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[2*x]^(3/2)*Sin[x],x]

[Out]

-(Sqrt[Cos[2*x]]*(-2*Cos[x] + Cos[3*x]))/8 - (3*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*
x]]])/(8*Sqrt[2])

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Maple [A]  time = 0.06, size = 55, normalized size = 1. \[{\frac{5\,\cos \left ( x \right ) }{8}\sqrt{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{2}\sqrt{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}}-{\frac{3\,\sqrt{2}}{16}\ln \left ( \cos \left ( x \right ) \sqrt{2}+\sqrt{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(2*x)^(3/2)*sin(x),x)

[Out]

5/8*cos(x)*(2*cos(x)^2-1)^(1/2)-1/2*cos(x)^3*(2*cos(x)^2-1)^(1/2)-3/16*ln(cos(x)
*2^(1/2)+(2*cos(x)^2-1)^(1/2))*2^(1/2)

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Maxima [A]  time = 2.28214, size = 1067, normalized size = 19.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*sqrt(2)*(4*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(((cos(4*x) -
2)*cos(1/2*arctan2(sin(4*x), cos(4*x))) + sin(4*x)*sin(1/2*arctan2(sin(4*x), co
s(4*x))) + cos(4*x) - 2)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) - (cos(1/2*arc
tan2(sin(4*x), cos(4*x)))*sin(4*x) - (cos(4*x) - 2)*sin(1/2*arctan2(sin(4*x), co
s(4*x))) - sin(4*x))*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))) + 3*log(sqrt(cos(
4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2
+ sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*sin(1/2*arctan2(sin(4*x), cos(
4*x) + 1))^2 + 2*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*cos(1/2*arctan
2(sin(4*x), cos(4*x) + 1)) + 1) - 3*log(sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x
) + 1)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 + sqrt(cos(4*x)^2 + sin(4*x)^2
+ 2*cos(4*x) + 1)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 - 2*(cos(4*x)^2 +
sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + 1)
+ 3*log(((cos(1/2*arctan2(sin(4*x), cos(4*x)))^2 + sin(1/2*arctan2(sin(4*x), co
s(4*x)))^2)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 + (cos(1/2*arctan2(sin(4*
x), cos(4*x)))^2 + sin(1/2*arctan2(sin(4*x), cos(4*x)))^2)*sin(1/2*arctan2(sin(4
*x), cos(4*x) + 1))^2)*sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1) + 2*(cos(4
*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(1/2*arctan2(sin(4*x), cos(4*x) +
1))*cos(1/2*arctan2(sin(4*x), cos(4*x))) + sin(1/2*arctan2(sin(4*x), cos(4*x) +
1))*sin(1/2*arctan2(sin(4*x), cos(4*x)))) + 1) - 3*log(((cos(1/2*arctan2(sin(4*
x), cos(4*x)))^2 + sin(1/2*arctan2(sin(4*x), cos(4*x)))^2)*cos(1/2*arctan2(sin(4
*x), cos(4*x) + 1))^2 + (cos(1/2*arctan2(sin(4*x), cos(4*x)))^2 + sin(1/2*arctan
2(sin(4*x), cos(4*x)))^2)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2)*sqrt(cos(4
*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1) - 2*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) +
1)^(1/4)*(cos(1/2*arctan2(sin(4*x), cos(4*x) + 1))*cos(1/2*arctan2(sin(4*x), co
s(4*x))) + sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))*sin(1/2*arctan2(sin(4*x), co
s(4*x)))) + 1))

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Fricas [A]  time = 0.243426, size = 142, normalized size = 2.58 \[ -\frac{1}{8} \,{\left (4 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1} + \frac{3}{128} \, \sqrt{2} \log \left (2048 \, \sqrt{2} \cos \left (x\right )^{8} - 2048 \, \sqrt{2} \cos \left (x\right )^{6} + 640 \, \sqrt{2} \cos \left (x\right )^{4} - 64 \, \sqrt{2} \cos \left (x\right )^{2} - 16 \,{\left (128 \, \cos \left (x\right )^{7} - 96 \, \cos \left (x\right )^{5} + 20 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1} + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*cos(x)^3 - 5*cos(x))*sqrt(2*cos(x)^2 - 1) + 3/128*sqrt(2)*log(2048*sqrt(
2)*cos(x)^8 - 2048*sqrt(2)*cos(x)^6 + 640*sqrt(2)*cos(x)^4 - 64*sqrt(2)*cos(x)^2
- 16*(128*cos(x)^7 - 96*cos(x)^5 + 20*cos(x)^3 - cos(x))*sqrt(2*cos(x)^2 - 1) +
sqrt(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)**(3/2)*sin(x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218005, size = 65, normalized size = 1.18 \[ -\frac{1}{8} \,{\left (4 \, \cos \left (x\right )^{2} - 5\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1} \cos \left (x\right ) + \frac{3}{16} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{2} \cos \left (x\right ) + \sqrt{2 \, \cos \left (x\right )^{2} - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*cos(x)^2 - 5)*sqrt(2*cos(x)^2 - 1)*cos(x) + 3/16*sqrt(2)*ln(abs(-sqrt(2)
*cos(x) + sqrt(2*cos(x)^2 - 1)))