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

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

[Out]

Sec[x]^3/3

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Rubi [A]  time = 0.0188348, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]^3*Tan[x],x]

[Out]

Sec[x]^3/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(tan(x)/cos(x)**3, x)

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Mathematica [A]  time = 0.00183606, size = 8, normalized size = 1. \[ \frac{\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]^3*Tan[x],x]

[Out]

Sec[x]^3/3

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Maple [A]  time = 0., size = 7, normalized size = 0.9 \[{\frac{ \left ( \sec \left ( x \right ) \right ) ^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sec(x)^3*tan(x),x)

[Out]

1/3*sec(x)^3

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Maxima [A]  time = 1.34065, size = 8, normalized size = 1. \[ \frac{1}{3 \, \cos \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/cos(x)^3

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Fricas [A]  time = 0.304967, size = 8, normalized size = 1. \[ \frac{1}{3 \, \cos \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/cos(x)^3

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Sympy [A]  time = 0.051899, size = 7, normalized size = 0.88 \[ \frac{1}{3 \cos ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sec(x)**3*tan(x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20016, size = 8, normalized size = 1. \[ \frac{1}{3 \, \cos \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/cos(x)^3