3.374 \(\int \cos (x) \cot (x) \sec (3 x) \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{2} \log \left (\csc ^2(x)-4\right ) \]

[Out]

-Log[-4 + Csc[x]^2]/2

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Rubi [A]  time = 0.0591418, antiderivative size = 17, normalized size of antiderivative = 1.55, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \log (\sin (x))-\frac{1}{2} \log \left (1-4 \sin ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Cot[x]*Sec[3*x],x]

[Out]

Log[Sin[x]] - Log[1 - 4*Sin[x]^2]/2

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Rubi in Sympy [A]  time = 21.4559, size = 20, normalized size = 1.82 \[ - \frac{\log{\left (- 4 \cos ^{2}{\left (x \right )} + 3 \right )}}{2} + \frac{\log{\left (- \cos ^{2}{\left (x \right )} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-4*cos(x)**2 + 3)/2 + log(-cos(x)**2 + 1)/2

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Mathematica [A]  time = 0.0188572, size = 17, normalized size = 1.55 \[ \log (\sin (x))-\frac{1}{2} \log (1-2 \cos (2 x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Cot[x]*Sec[3*x],x]

[Out]

-Log[1 - 2*Cos[2*x]]/2 + Log[Sin[x]]

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Maple [B]  time = 0.059, size = 27, normalized size = 2.5 \[{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{2}}+{\frac{\ln \left ( \cos \left ( x \right ) -1 \right ) }{2}}-{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^2/cos(3*x)/sin(x),x)

[Out]

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

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Maxima [A]  time = 1.34871, size = 124, normalized size = 11.27 \[ -\frac{1}{4} \, \log \left (-2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} - 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(-2*(cos(2*x) - 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 + sin(4*x)^2 - 2*s
in(4*x)*sin(2*x) + sin(2*x)^2 - 2*cos(2*x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 +
2*cos(x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [A]  time = 0.22399, size = 23, normalized size = 2.09 \[ -\frac{1}{2} \, \log \left (4 \, \cos \left (x\right )^{2} - 3\right ) + \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(cos(x)**2/(sin(x)*cos(3*x)), x)

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GIAC/XCAS [A]  time = 0.224393, size = 32, normalized size = 2.91 \[ \frac{1}{2} \,{\rm ln}\left (-\cos \left (x\right )^{2} + 1\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | 4 \, \cos \left (x\right )^{2} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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