3.367 \(\int \cos (4 x) \sec (x) \, dx\)

Optimal. Leaf size=12 \[ \tanh ^{-1}(\sin (x))-\frac{8 \sin ^3(x)}{3} \]

[Out]

ArcTanh[Sin[x]] - (8*Sin[x]^3)/3

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Rubi [A]  time = 0.0431068, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \tanh ^{-1}(\sin (x))-\frac{8 \sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Cos[4*x]*Sec[x],x]

[Out]

ArcTanh[Sin[x]] - (8*Sin[x]^3)/3

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Rubi in Sympy [A]  time = 1.8727, size = 15, normalized size = 1.25 \[ x \cos{\left (3 x \right )} + \log{\left (\cos{\left (x \right )} \right )} \sin{\left (3 x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(4*x)/cos(x),x)

[Out]

x*cos(3*x) + log(cos(x))*sin(3*x)

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Mathematica [B]  time = 0.0141269, size = 45, normalized size = 3.75 \[ -2 \sin (x)+\frac{2}{3} \sin (3 x)-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[4*x]*Sec[x],x]

[Out]

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

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Maple [B]  time = 0.027, size = 22, normalized size = 1.8 \[ \ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) +{\frac{ \left ( 16+8\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}}-8\,\sin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(4*x)/cos(x),x)

[Out]

ln(sec(x)+tan(x))+8/3*(2+cos(x)^2)*sin(x)-8*sin(x)

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Maxima [A]  time = 1.47201, size = 28, normalized size = 2.33 \[ -\frac{8}{3} \, \sin \left (x\right )^{3} + \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (\sin \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)/cos(x),x, algorithm="maxima")

[Out]

-8/3*sin(x)^3 + 1/2*log(sin(x) + 1) - 1/2*log(sin(x) - 1)

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Fricas [A]  time = 0.233364, size = 36, normalized size = 3. \[ \frac{8}{3} \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)/cos(x),x, algorithm="fricas")

[Out]

8/3*(cos(x)^2 - 1)*sin(x) + 1/2*log(sin(x) + 1) - 1/2*log(-sin(x) + 1)

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Sympy [A]  time = 0.842551, size = 24, normalized size = 2. \[ - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{2} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{2} - \frac{8 \sin ^{3}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)/cos(x),x)

[Out]

-log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - 8*sin(x)**3/3

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GIAC/XCAS [A]  time = 0.204612, size = 31, normalized size = 2.58 \[ -\frac{8}{3} \, \sin \left (x\right )^{3} + \frac{1}{2} \,{\rm ln}\left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \,{\rm ln}\left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)/cos(x),x, algorithm="giac")

[Out]

-8/3*sin(x)^3 + 1/2*ln(sin(x) + 1) - 1/2*ln(-sin(x) + 1)