3.43 \(\int \tan ^4(x) \, dx\)

Optimal. Leaf size=14 \[ x+\frac{\tan ^3(x)}{3}-\tan (x) \]

[Out]

x - Tan[x] + Tan[x]^3/3

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Rubi [A]  time = 0.0159044, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ x+\frac{\tan ^3(x)}{3}-\tan (x) \]

Antiderivative was successfully verified.

[In]  Int[Tan[x]^4,x]

[Out]

x - Tan[x] + Tan[x]^3/3

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Rubi in Sympy [A]  time = 0.473357, size = 10, normalized size = 0.71 \[ x + \frac{\tan ^{3}{\left (x \right )}}{3} - \tan{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(tan(x)**4,x)

[Out]

x + tan(x)**3/3 - tan(x)

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Mathematica [A]  time = 0.0051962, size = 18, normalized size = 1.29 \[ x-\frac{4 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Tan[x]^4,x]

[Out]

x - (4*Tan[x])/3 + (Sec[x]^2*Tan[x])/3

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Maple [A]  time = 0.004, size = 13, normalized size = 0.9 \[ x-\tan \left ( x \right ) +{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tan(x)^4,x)

[Out]

x-tan(x)+1/3*tan(x)^3

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Maxima [A]  time = 1.49914, size = 16, normalized size = 1.14 \[ \frac{1}{3} \, \tan \left (x\right )^{3} + x - \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*tan(x)^3 + x - tan(x)

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Fricas [A]  time = 0.242919, size = 16, normalized size = 1.14 \[ \frac{1}{3} \, \tan \left (x\right )^{3} + x - \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*tan(x)^3 + x - tan(x)

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Sympy [A]  time = 0.049686, size = 19, normalized size = 1.36 \[ x + \frac{\sin ^{3}{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} - \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.232812, size = 16, normalized size = 1.14 \[ \frac{1}{3} \, \tan \left (x\right )^{3} + x - \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*tan(x)^3 + x - tan(x)