3.114 \(\int \sin ^2(x) \tan (x) \, dx\)

Optimal. Leaf size=14 \[ \frac{\cos ^2(x)}{2}-\log (\cos (x)) \]

[Out]

Cos[x]^2/2 - Log[Cos[x]]

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Rubi [A]  time = 0.0219364, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\cos ^2(x)}{2}-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]^2*Tan[x],x]

[Out]

Cos[x]^2/2 - Log[Cos[x]]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**2*tan(x),x)

[Out]

-log(cos(x)) + Integral(x, (x, cos(x)))

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Mathematica [A]  time = 0.00692507, size = 14, normalized size = 1. \[ \frac{1}{4} \cos (2 x)-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]^2*Tan[x],x]

[Out]

Cos[2*x]/4 - Log[Cos[x]]

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Maple [A]  time = 0.014, size = 13, normalized size = 0.9 \[ -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{2}}-\ln \left ( \cos \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^2*tan(x),x)

[Out]

-1/2*sin(x)^2-ln(cos(x))

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Maxima [A]  time = 1.33534, size = 22, normalized size = 1.57 \[ -\frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sin(x)^2 - 1/2*log(sin(x)^2 - 1)

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Fricas [A]  time = 0.264833, size = 19, normalized size = 1.36 \[ \frac{1}{2} \, \cos \left (x\right )^{2} - \log \left (-\cos \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*cos(x)^2 - log(-cos(x))

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Sympy [A]  time = 0.07138, size = 10, normalized size = 0.71 \[ - \log{\left (\cos{\left (x \right )} \right )} + \frac{\cos ^{2}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**2*tan(x),x)

[Out]

-log(cos(x)) + cos(x)**2/2

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GIAC/XCAS [A]  time = 0.201228, size = 24, normalized size = 1.71 \[ -\frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2} \,{\rm ln}\left (-\sin \left (x\right )^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sin(x)^2 - 1/2*ln(-sin(x)^2 + 1)