3.90 \(\int \tan ^5(x) \, dx\)

Optimal. Leaf size=22 \[ \frac{\tan ^4(x)}{4}-\frac{\tan ^2(x)}{2}-\log (\cos (x)) \]

[Out]

-Log[Cos[x]] - Tan[x]^2/2 + Tan[x]^4/4

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Rubi [A]  time = 0.0101662, antiderivative size = 22, 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, 3475} \[ \frac{\tan ^4(x)}{4}-\frac{\tan ^2(x)}{2}-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^5,x]

[Out]

-Log[Cos[x]] - Tan[x]^2/2 + 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 \tan ^5(x) \, dx &=\frac{\tan ^4(x)}{4}-\int \tan ^3(x) \, dx\\ &=-\frac{1}{2} \tan ^2(x)+\frac{\tan ^4(x)}{4}+\int \tan (x) \, dx\\ &=-\log (\cos (x))-\frac{\tan ^2(x)}{2}+\frac{\tan ^4(x)}{4}\\ \end{align*}

Mathematica [A]  time = 0.0030332, size = 20, normalized size = 0.91 \[ \frac{\sec ^4(x)}{4}-\sec ^2(x)-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^5,x]

[Out]

-Log[Cos[x]] - Sec[x]^2 + Sec[x]^4/4

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Maple [A]  time = 0.002, size = 23, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{4}}{4}}+{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^5,x)

[Out]

-1/2*tan(x)^2+1/4*tan(x)^4+1/2*ln(tan(x)^2+1)

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Maxima [A]  time = 0.925153, size = 46, normalized size = 2.09 \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} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.2591, size = 77, normalized size = 3.5 \begin{align*} \frac{1}{4} \, \tan \left (x\right )^{4} - \frac{1}{2} \, \tan \left (x\right )^{2} - \frac{1}{2} \, \log \left (\frac{1}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*tan(x)^4 - 1/2*tan(x)^2 - 1/2*log(1/(tan(x)^2 + 1))

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Sympy [A]  time = 0.103746, size = 20, normalized size = 0.91 \begin{align*} - \frac{4 \cos ^{2}{\left (x \right )} - 1}{4 \cos ^{4}{\left (x \right )}} - \log{\left (\cos{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05882, size = 30, normalized size = 1.36 \begin{align*} \frac{1}{4} \, \tan \left (x\right )^{4} - \frac{1}{2} \, \tan \left (x\right )^{2} + \frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*tan(x)^4 - 1/2*tan(x)^2 + 1/2*log(tan(x)^2 + 1)