3.74 \(\int \tan ^4(y) \, dy\)

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

[Out]

y - Tan[y] + Tan[y]^3/3

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Rubi [A]  time = 0.014388, 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 \[ y+\frac{\tan ^3(y)}{3}-\tan (y) \]

Antiderivative was successfully verified.

[In]  Int[Tan[y]^4,y]

[Out]

y - Tan[y] + Tan[y]^3/3

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Rubi in Sympy [A]  time = 2.58295, size = 19, normalized size = 1.36 \[ y + \frac{\sin ^{3}{\left (y \right )}}{3 \cos ^{3}{\left (y \right )}} - \frac{\sin{\left (y \right )}}{\cos{\left (y \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(y)**4/cos(y)**4,y)

[Out]

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

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Mathematica [A]  time = 0.0104996, size = 18, normalized size = 1.29 \[ y-\frac{4 \tan (y)}{3}+\frac{1}{3} \tan (y) \sec ^2(y) \]

Antiderivative was successfully verified.

[In]  Integrate[Tan[y]^4,y]

[Out]

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

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Maple [A]  time = 0.011, size = 13, normalized size = 0.9 \[ y-\tan \left ( y \right ) +{\frac{ \left ( \tan \left ( y \right ) \right ) ^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(y)^4/cos(y)^4,y)

[Out]

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

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Maxima [A]  time = 1.50277, size = 16, normalized size = 1.14 \[ \frac{1}{3} \, \tan \left (y\right )^{3} + y - \tan \left (y\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(y)^4/cos(y)^4,y, algorithm="maxima")

[Out]

1/3*tan(y)^3 + y - tan(y)

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Fricas [A]  time = 0.229481, size = 35, normalized size = 2.5 \[ \frac{3 \, y \cos \left (y\right )^{3} -{\left (4 \, \cos \left (y\right )^{2} - 1\right )} \sin \left (y\right )}{3 \, \cos \left (y\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(y)^4/cos(y)^4,y, algorithm="fricas")

[Out]

1/3*(3*y*cos(y)^3 - (4*cos(y)^2 - 1)*sin(y))/cos(y)^3

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Sympy [A]  time = 0.047141, size = 19, normalized size = 1.36 \[ y + \frac{\sin ^{3}{\left (y \right )}}{3 \cos ^{3}{\left (y \right )}} - \frac{\sin{\left (y \right )}}{\cos{\left (y \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(y)**4/cos(y)**4,y)

[Out]

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

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GIAC/XCAS [A]  time = 0.207196, size = 16, normalized size = 1.14 \[ \frac{1}{3} \, \tan \left (y\right )^{3} + y - \tan \left (y\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(y)^4/cos(y)^4,y, algorithm="giac")

[Out]

1/3*tan(y)^3 + y - tan(y)