3.385 \(\int \sec (2 x) \sin ^2(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \tanh ^{-1}(2 \sin (x) \cos (x))-\frac{x}{2} \]

[Out]

-x/2 + ArcTanh[2*Cos[x]*Sin[x]]/4

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Rubi [A]  time = 0.0685826, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{4} \tanh ^{-1}(2 \sin (x) \cos (x))-\frac{x}{2} \]

Antiderivative was successfully verified.

[In]  Int[Sec[2*x]*Sin[x]^2,x]

[Out]

-x/2 + ArcTanh[2*Cos[x]*Sin[x]]/4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sin(x)**2/cos(2*x), x)

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Mathematica [A]  time = 0.0111216, size = 28, normalized size = 1.65 \[ -\frac{x}{2}-\frac{1}{4} \log (\cos (x)-\sin (x))+\frac{1}{4} \log (\sin (x)+\cos (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[2*x]*Sin[x]^2,x]

[Out]

-x/2 - Log[Cos[x] - Sin[x]]/4 + Log[Cos[x] + Sin[x]]/4

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Maple [A]  time = 0.051, size = 19, normalized size = 1.1 \[{\frac{\ln \left ( 1+\tan \left ( x \right ) \right ) }{4}}-{\frac{\ln \left ( -1+\tan \left ( x \right ) \right ) }{4}}-{\frac{x}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^2/cos(2*x),x)

[Out]

1/4*ln(1+tan(x))-1/4*ln(-1+tan(x))-1/2*x

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Maxima [A]  time = 1.52528, size = 173, normalized size = 10.18 \[ -\frac{1}{2} \, x - \frac{1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) + \frac{1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) + \frac{1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256314, size = 35, normalized size = 2.06 \[ -\frac{1}{2} \, x + \frac{1}{8} \, \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - \frac{1}{8} \, \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.80675, size = 22, normalized size = 1.29 \[ - \frac{x}{2} - \frac{\log{\left (\sin{\left (2 x \right )} - 1 \right )}}{8} + \frac{\log{\left (\sin{\left (2 x \right )} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**2/cos(2*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226357, size = 27, normalized size = 1.59 \[ -\frac{1}{2} \, x + \frac{1}{4} \,{\rm ln}\left ({\left | \tan \left (x\right ) + 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | \tan \left (x\right ) - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x + 1/4*ln(abs(tan(x) + 1)) - 1/4*ln(abs(tan(x) - 1))