3.95 \(\int \sec ^6(x) \tan ^3(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\sec ^8(x)}{8}-\frac{\sec ^6(x)}{6} \]

[Out]

-Sec[x]^6/6 + Sec[x]^8/8

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Rubi [A]  time = 0.0368012, 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 \[ \frac{\sec ^8(x)}{8}-\frac{\sec ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^6*Tan[x]^3,x]

[Out]

-Sec[x]^6/6 + Sec[x]^8/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan ^{3}{\left (x \right )}}{\cos ^{6}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sec(x)**6*tan(x)**3,x)

[Out]

Integral(tan(x)**3/cos(x)**6, x)

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Mathematica [A]  time = 0.0104471, size = 17, normalized size = 1. \[ \frac{\sec ^8(x)}{8}-\frac{\sec ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^6*Tan[x]^3,x]

[Out]

-Sec[x]^6/6 + Sec[x]^8/8

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Maple [B]  time = 0.02, size = 32, normalized size = 1.9 \[{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{8\, \left ( \cos \left ( x \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{12\, \left ( \cos \left ( x \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{24\, \left ( \cos \left ( x \right ) \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^6*tan(x)^3,x)

[Out]

1/8*sin(x)^4/cos(x)^8+1/12*sin(x)^4/cos(x)^6+1/24*sin(x)^4/cos(x)^4

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Maxima [A]  time = 1.37368, size = 49, normalized size = 2.88 \[ \frac{4 \, \sin \left (x\right )^{2} - 1}{24 \,{\left (\sin \left (x\right )^{8} - 4 \, \sin \left (x\right )^{6} + 6 \, \sin \left (x\right )^{4} - 4 \, \sin \left (x\right )^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^6*tan(x)^3,x, algorithm="maxima")

[Out]

1/24*(4*sin(x)^2 - 1)/(sin(x)^8 - 4*sin(x)^6 + 6*sin(x)^4 - 4*sin(x)^2 + 1)

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Fricas [A]  time = 0.25274, size = 19, normalized size = 1.12 \[ -\frac{4 \, \cos \left (x\right )^{2} - 3}{24 \, \cos \left (x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^6*tan(x)^3,x, algorithm="fricas")

[Out]

-1/24*(4*cos(x)^2 - 3)/cos(x)^8

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Sympy [A]  time = 0.132363, size = 15, normalized size = 0.88 \[ - \frac{4 \cos ^{2}{\left (x \right )} - 3}{24 \cos ^{8}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**6*tan(x)**3,x)

[Out]

-(4*cos(x)**2 - 3)/(24*cos(x)**8)

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GIAC/XCAS [A]  time = 0.204358, size = 19, normalized size = 1.12 \[ -\frac{4 \, \cos \left (x\right )^{2} - 3}{24 \, \cos \left (x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)^6*tan(x)^3,x, algorithm="giac")

[Out]

-1/24*(4*cos(x)^2 - 3)/cos(x)^8