3.354 \(\int \sec ^2(x) \tan ^2(x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\tan ^3(x)}{3} \]

[Out]

Tan[x]^3/3

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Rubi [A]  time = 0.0324581, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^2*Tan[x]^2,x]

[Out]

Tan[x]^3/3

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Rubi in Sympy [A]  time = 1.93293, size = 5, normalized size = 0.62 \[ \frac{\tan ^{3}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

tan(x)**3/3

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Mathematica [A]  time = 0.00380236, size = 8, normalized size = 1. \[ \frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^2*Tan[x]^2,x]

[Out]

Tan[x]^3/3

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Maple [A]  time = 0.013, size = 11, normalized size = 1.4 \[{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3\, \left ( \cos \left ( x \right ) \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^2*tan(x)^2,x)

[Out]

1/3*sin(x)^3/cos(x)^3

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Maxima [A]  time = 1.43849, size = 8, normalized size = 1. \[ \frac{1}{3} \, \tan \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*tan(x)^3

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Fricas [A]  time = 0.215282, size = 19, normalized size = 2.38 \[ -\frac{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(cos(x)^2 - 1)*sin(x)/cos(x)^3

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Sympy [A]  time = 0.048051, size = 17, normalized size = 2.12 \[ - \frac{\sin{\left (x \right )}}{3 \cos{\left (x \right )}} + \frac{\sin{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**2*tan(x)**2,x)

[Out]

-sin(x)/(3*cos(x)) + sin(x)/(3*cos(x)**3)

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GIAC/XCAS [A]  time = 0.199104, size = 8, normalized size = 1. \[ \frac{1}{3} \, \tan \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*tan(x)^3