3.1.74 \(\int \tan ^4(y) \, dy\) [74]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3554, 8} \begin {gather*} y+\frac {\tan ^3(y)}{3}-\tan (y) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[y]^4,y]

[Out]

y - Tan[y] + Tan[y]^3/3

Rule 8

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

Rule 3554

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]

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 = 0.01, size = 18, normalized size = 1.29 \begin {gather*} y-\frac {4 \tan (y)}{3}+\frac {1}{3} \sec ^2(y) \tan (y) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[y]^4,y]

[Out]

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

________________________________________________________________________________________

Mathics [A]
time = 1.75, size = 12, normalized size = 0.86 \begin {gather*} y-\text {Tan}\left [y\right ]+\frac {\text {Tan}\left [y\right ]^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[y]^4/Cos[y]^4,y]')

[Out]

y - Tan[y] + Tan[y] ^ 3 / 3

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 13, normalized size = 0.93

method	result	size
default	\(y -\tan \left (y \right )+\frac {\left (\tan ^{3}\left (y \right )\right )}{3}\)	\(13\)
risch	\(y -\frac {4 i \left (3 \,{\mathrm e}^{4 i y}+3 \,{\mathrm e}^{2 i y}+2\right )}{3 \left ({\mathrm e}^{2 i y}+1\right )^{3}}\)	\(31\)
norman	\(\frac {y \left (\tan ^{6}\left (\frac {y}{2}\right )\right )-y -\frac {20 \left (\tan ^{3}\left (\frac {y}{2}\right )\right )}{3}+2 \left (\tan ^{5}\left (\frac {y}{2}\right )\right )+3 y \left (\tan ^{2}\left (\frac {y}{2}\right )\right )-3 y \left (\tan ^{4}\left (\frac {y}{2}\right )\right )+2 \tan \left (\frac {y}{2}\right )}{\left (\tan ^{2}\left (\frac {y}{2}\right )-1\right )^{3}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)^4/cos(y)^4,y,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.34, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, \tan \left (y\right )^{3} + y - \tan \left (y\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.31, size = 26, normalized size = 1.86 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 19, normalized size = 1.36 \begin {gather*} y + \frac {\sin ^{3}{\left (y \right )}}{3 \cos ^{3}{\left (y \right )}} - \frac {\sin {\left (y \right )}}{\cos {\left (y \right )}} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 24, normalized size = 1.71 \begin {gather*} 2 \left (\frac {\frac {4}{3} \tan ^{3}y-4 \tan y}{8}+\frac {y}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)^4/cos(y)^4,y)

[Out]

1/3*tan(y)^3 + y - tan(y)

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\mathrm {tan}\left (y\right )}^3}{3}-\mathrm {tan}\left (y\right )+y \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)^4/cos(y)^4,y)

[Out]

y - tan(y) + tan(y)^3/3

________________________________________________________________________________________