3.394 \(\int \frac{1}{(1-\cos (3 x))^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{\sin (3 x)}{6 (1-\cos (3 x))^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sin (3 x)}{\sqrt{2} \sqrt{1-\cos (3 x)}}\right )}{6 \sqrt{2}} \]

[Out]

-ArcTanh[Sin[3*x]/(Sqrt[2]*Sqrt[1 - Cos[3*x]])]/(6*Sqrt[2]) - Sin[3*x]/(6*(1 - C
os[3*x])^(3/2))

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Rubi [A]  time = 0.0494847, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sin (3 x)}{6 (1-\cos (3 x))^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sin (3 x)}{\sqrt{2} \sqrt{1-\cos (3 x)}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - Cos[3*x])^(-3/2),x]

[Out]

-ArcTanh[Sin[3*x]/(Sqrt[2]*Sqrt[1 - Cos[3*x]])]/(6*Sqrt[2]) - Sin[3*x]/(6*(1 - C
os[3*x])^(3/2))

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Rubi in Sympy [A]  time = 0.79686, size = 48, normalized size = 0.91 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sin{\left (3 x \right )}}{2 \sqrt{- \cos{\left (3 x \right )} + 1}} \right )}}{12} - \frac{\sin{\left (3 x \right )}}{6 \left (- \cos{\left (3 x \right )} + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*atanh(sqrt(2)*sin(3*x)/(2*sqrt(-cos(3*x) + 1)))/12 - sin(3*x)/(6*(-cos(
3*x) + 1)**(3/2))

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Mathematica [A]  time = 0.185542, size = 61, normalized size = 1.15 \[ -\frac{\sin ^3\left (\frac{3 x}{2}\right ) \left (\csc ^2\left (\frac{3 x}{4}\right )-\sec ^2\left (\frac{3 x}{4}\right )-4 \log \left (\sin \left (\frac{3 x}{4}\right )\right )+4 \log \left (\cos \left (\frac{3 x}{4}\right )\right )\right )}{12 (1-\cos (3 x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - Cos[3*x])^(-3/2),x]

[Out]

-((Csc[(3*x)/4]^2 + 4*Log[Cos[(3*x)/4]] - 4*Log[Sin[(3*x)/4]] - Sec[(3*x)/4]^2)*
Sin[(3*x)/2]^3)/(12*(1 - Cos[3*x])^(3/2))

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Maple [A]  time = 0.081, size = 52, normalized size = 1. \[ -{\frac{\sqrt{2}}{6} \left ({\frac{1}{2}\cos \left ({\frac{3\,x}{2}} \right ) }+{\frac{1}{4} \left ( \ln \left ( 1+\cos \left ({\frac{3\,x}{2}} \right ) \right ) -\ln \left ( \cos \left ({\frac{3\,x}{2}} \right ) -1 \right ) \right ) \left ( \sin \left ({\frac{3\,x}{2}} \right ) \right ) ^{2}} \right ) \left ( \sin \left ({\frac{3\,x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{ \left ( \sin \left ({\frac{3\,x}{2}} \right ) \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-cos(3*x))^(3/2),x)

[Out]

-1/6*(1/2*cos(3/2*x)+1/4*(ln(1+cos(3/2*x))-ln(cos(3/2*x)-1))*sin(3/2*x)^2)/sin(3
/2*x)*2^(1/2)/(sin(3/2*x)^2)^(1/2)

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Maxima [A]  time = 1.6159, size = 103, normalized size = 1.94 \[ -\frac{1}{12} \, \sqrt{2}{\left (\frac{{\left (\frac{\sin \left (3 \, x\right )^{2}}{{\left (\cos \left (3 \, x\right ) + 1\right )}^{2}} + 1\right )}^{\frac{3}{2}}{\left (\cos \left (3 \, x\right ) + 1\right )}^{2}}{\sin \left (3 \, x\right )^{2}} - \sqrt{\frac{\sin \left (3 \, x\right )^{2}}{{\left (\cos \left (3 \, x\right ) + 1\right )}^{2}} + 1} + \operatorname{arsinh}\left (\frac{\cos \left (3 \, x\right ) + 1}{{\left | \sin \left (3 \, x\right ) \right |}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-cos(3*x) + 1)^(-3/2),x, algorithm="maxima")

[Out]

-1/12*sqrt(2)*((sin(3*x)^2/(cos(3*x) + 1)^2 + 1)^(3/2)*(cos(3*x) + 1)^2/sin(3*x)
^2 - sqrt(sin(3*x)^2/(cos(3*x) + 1)^2 + 1) + arcsinh((cos(3*x) + 1)/abs(sin(3*x)
)))

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Fricas [A]  time = 0.215972, size = 144, normalized size = 2.72 \[ \frac{{\left (\sqrt{2} \cos \left (3 \, x\right ) - \sqrt{2}\right )} \log \left (-\frac{{\left (\sqrt{2} \cos \left (3 \, x\right ) + 3 \, \sqrt{2}\right )} \sqrt{-\cos \left (3 \, x\right ) + 1} - 4 \, \sin \left (3 \, x\right )}{{\left (\cos \left (3 \, x\right ) - 1\right )} \sqrt{-\cos \left (3 \, x\right ) + 1}}\right ) \sin \left (3 \, x\right ) + 4 \,{\left (\cos \left (3 \, x\right ) + 1\right )} \sqrt{-\cos \left (3 \, x\right ) + 1}}{24 \,{\left (\cos \left (3 \, x\right ) - 1\right )} \sin \left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-cos(3*x) + 1)^(-3/2),x, algorithm="fricas")

[Out]

1/24*((sqrt(2)*cos(3*x) - sqrt(2))*log(-((sqrt(2)*cos(3*x) + 3*sqrt(2))*sqrt(-co
s(3*x) + 1) - 4*sin(3*x))/((cos(3*x) - 1)*sqrt(-cos(3*x) + 1)))*sin(3*x) + 4*(co
s(3*x) + 1)*sqrt(-cos(3*x) + 1))/((cos(3*x) - 1)*sin(3*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- \cos{\left (3 x \right )} + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-cos(3*x) + 1)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.256494, size = 80, normalized size = 1.51 \[ -\frac{\sqrt{2}{\left (\frac{2 \, \sqrt{\tan \left (\frac{3}{2} \, x\right )^{2} + 1}}{\tan \left (\frac{3}{2} \, x\right )^{2}} +{\rm ln}\left (\sqrt{\tan \left (\frac{3}{2} \, x\right )^{2} + 1} + 1\right ) -{\rm ln}\left (\sqrt{\tan \left (\frac{3}{2} \, x\right )^{2} + 1} - 1\right )\right )}}{24 \,{\rm sign}\left (\tan \left (\frac{3}{2} \, x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-cos(3*x) + 1)^(-3/2),x, algorithm="giac")

[Out]

-1/24*sqrt(2)*(2*sqrt(tan(3/2*x)^2 + 1)/tan(3/2*x)^2 + ln(sqrt(tan(3/2*x)^2 + 1)
+ 1) - ln(sqrt(tan(3/2*x)^2 + 1) - 1))/sign(tan(3/2*x))