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 = 0.0094383, 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, Rules used = {3473, 8} \[ y+\frac{\tan ^3(y)}{3}-\tan (y) \]

Antiderivative was successfully verified.

[In]

Int[Tan[y]^4,y]

[Out]

y - Tan[y] + Tan[y]^3/3

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \tan ^4(y) \, dy &=\frac{\tan ^3(y)}{3}-\int \tan ^2(y) \, dy\\ &=-\tan (y)+\frac{\tan ^3(y)}{3}+\int 1 \, dy\\ &=y-\tan (y)+\frac{\tan ^3(y)}{3}\\ \end{align*}

Mathematica [A]  time = 0.006699, 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 = 0.007, size = 13, normalized size = 0.9 \begin{align*} y-\tan \left ( y \right ) +{\frac{ \left ( \tan \left ( y \right ) \right ) ^{3}}{3}} \end{align*}

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 = 1.41807, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{3} \, \tan \left (y\right )^{3} + y - \tan \left (y\right ) \end{align*}

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 [B]  time = 2.2519, size = 74, normalized size = 5.29 \begin{align*} \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}} \end{align*}

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 = 0.060681, size = 19, normalized size = 1.36 \begin{align*} y + \frac{\sin ^{3}{\left (y \right )}}{3 \cos ^{3}{\left (y \right )}} - \frac{\sin{\left (y \right )}}{\cos{\left (y \right )}} \end{align*}

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 [A]  time = 1.06633, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{3} \, \tan \left (y\right )^{3} + y - \tan \left (y\right ) \end{align*}

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)