3.117 \(\int \sec ^3(x) \tan ^3(x) \, dx\)

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

[Out]

-Sec[x]^3/3 + Sec[x]^5/5

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Rubi [A]  time = 0.0376297, 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 \[ \frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^3*Tan[x]^3,x]

[Out]

-Sec[x]^3/3 + Sec[x]^5/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan ^{3}{\left (x \right )}}{\cos ^{3}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)**3*tan(x)**3,x)

[Out]

Integral(tan(x)**3/cos(x)**3, x)

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Mathematica [A]  time = 0.0115383, size = 17, normalized size = 1. \[ \frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^3*Tan[x]^3,x]

[Out]

-Sec[x]^3/3 + Sec[x]^5/5

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Maple [B]  time = 0., size = 42, normalized size = 2.5 \[{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{15\, \left ( \cos \left ( x \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{15\,\cos \left ( x \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^3*tan(x)^3,x)

[Out]

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

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Maxima [A]  time = 1.35367, size = 19, normalized size = 1.12 \[ -\frac{5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^3*tan(x)^3,x, algorithm="maxima")

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5

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Fricas [A]  time = 0.214097, size = 19, normalized size = 1.12 \[ -\frac{5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^3*tan(x)^3,x, algorithm="fricas")

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5

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Sympy [A]  time = 0.108544, size = 15, normalized size = 0.88 \[ - \frac{5 \cos ^{2}{\left (x \right )} - 3}{15 \cos ^{5}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**3*tan(x)**3,x)

[Out]

-(5*cos(x)**2 - 3)/(15*cos(x)**5)

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GIAC/XCAS [A]  time = 0.200934, size = 19, normalized size = 1.12 \[ -\frac{5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^3*tan(x)^3,x, algorithm="giac")

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5