3.841 \(\int (\tan ^3(x)+\tan ^5(x)) \, dx\)

Optimal. Leaf size=8 \[ \frac{\tan ^4(x)}{4} \]

[Out]

Tan[x]^4/4

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Rubi [A]  time = 0.0167934, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3473, 3475} \[ \frac{\tan ^4(x)}{4} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^3 + Tan[x]^5,x]

[Out]

Tan[x]^4/4

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (\tan ^3(x)+\tan ^5(x)\right ) \, dx &=\int \tan ^3(x) \, dx+\int \tan ^5(x) \, dx\\ &=\frac{\tan ^2(x)}{2}+\frac{\tan ^4(x)}{4}-\int \tan (x) \, dx-\int \tan ^3(x) \, dx\\ &=\log (\cos (x))+\frac{\tan ^4(x)}{4}+\int \tan (x) \, dx\\ &=\frac{\tan ^4(x)}{4}\\ \end{align*}

Mathematica [A]  time = 0.0048798, size = 8, normalized size = 1. \[ \frac{\tan ^4(x)}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^3 + Tan[x]^5,x]

[Out]

Tan[x]^4/4

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Maple [A]  time = 0.004, size = 7, normalized size = 0.9 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^3+tan(x)^5,x)

[Out]

1/4*tan(x)^4

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Maxima [B]  time = 0.948578, size = 47, normalized size = 5.88 \begin{align*} \frac{4 \, \sin \left (x\right )^{2} - 3}{4 \,{\left (\sin \left (x\right )^{4} - 2 \, \sin \left (x\right )^{2} + 1\right )}} - \frac{1}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3+tan(x)^5,x, algorithm="maxima")

[Out]

1/4*(4*sin(x)^2 - 3)/(sin(x)^4 - 2*sin(x)^2 + 1) - 1/2/(sin(x)^2 - 1)

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Fricas [A]  time = 1.92514, size = 19, normalized size = 2.38 \begin{align*} \frac{1}{4} \, \tan \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3+tan(x)^5,x, algorithm="fricas")

[Out]

1/4*tan(x)^4

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Sympy [B]  time = 0.125427, size = 22, normalized size = 2.75 \begin{align*} - \frac{4 \cos ^{2}{\left (x \right )} - 1}{4 \cos ^{4}{\left (x \right )}} + \frac{1}{2 \cos ^{2}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**3+tan(x)**5,x)

[Out]

-(4*cos(x)**2 - 1)/(4*cos(x)**4) + 1/(2*cos(x)**2)

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Giac [A]  time = 1.07061, size = 8, normalized size = 1. \begin{align*} \frac{1}{4} \, \tan \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3+tan(x)^5,x, algorithm="giac")

[Out]

1/4*tan(x)^4