3.87 \(\int \sec ^4(x) \tan ^2(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\tan ^5(x)}{5}+\frac{\tan ^3(x)}{3} \]

[Out]

Tan[x]^3/3 + Tan[x]^5/5

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Rubi [A]  time = 0.0363302, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tan ^5(x)}{5}+\frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^4*Tan[x]^2,x]

[Out]

Tan[x]^3/3 + Tan[x]^5/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan ^{2}{\left (x \right )}}{\cos ^{4}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)**4*tan(x)**2,x)

[Out]

Integral(tan(x)**2/cos(x)**4, x)

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Mathematica [A]  time = 0.0172234, size = 27, normalized size = 1.59 \[ -\frac{2 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)-\frac{1}{15} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^4*Tan[x]^2,x]

[Out]

(-2*Tan[x])/15 - (Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5

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Maple [A]  time = 0.022, size = 22, normalized size = 1.3 \[{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{15\, \left ( \cos \left ( x \right ) \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^4*tan(x)^2,x)

[Out]

1/5*sin(x)^3/cos(x)^5+2/15*sin(x)^3/cos(x)^3

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Maxima [A]  time = 1.34295, size = 18, normalized size = 1.06 \[ \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{1}{3} \, \tan \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^4*tan(x)^2,x, algorithm="maxima")

[Out]

1/5*tan(x)^5 + 1/3*tan(x)^3

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Fricas [A]  time = 0.211663, size = 27, normalized size = 1.59 \[ -\frac{{\left (2 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 3\right )} \sin \left (x\right )}{15 \, \cos \left (x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^4*tan(x)^2,x, algorithm="fricas")

[Out]

-1/15*(2*cos(x)^4 + cos(x)^2 - 3)*sin(x)/cos(x)^5

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Sympy [A]  time = 0.053007, size = 29, normalized size = 1.71 \[ - \frac{2 \sin{\left (x \right )}}{15 \cos{\left (x \right )}} - \frac{\sin{\left (x \right )}}{15 \cos ^{3}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**4*tan(x)**2,x)

[Out]

-2*sin(x)/(15*cos(x)) - sin(x)/(15*cos(x)**3) + sin(x)/(5*cos(x)**5)

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GIAC/XCAS [A]  time = 0.199124, size = 18, normalized size = 1.06 \[ \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{1}{3} \, \tan \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^4*tan(x)^2,x, algorithm="giac")

[Out]

1/5*tan(x)^5 + 1/3*tan(x)^3