### 3.375 $$\int \tan ^3(x) \, dx$$

Optimal. Leaf size=12 $\frac{\tan ^2(x)}{2}+\log (\cos (x))$

[Out]

Log[Cos[x]] + Tan[x]^2/2

________________________________________________________________________________________

Rubi [A]  time = 0.0063899, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3473, 3475} $\frac{\tan ^2(x)}{2}+\log (\cos (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Tan[x]^3,x]

[Out]

Log[Cos[x]] + Tan[x]^2/2

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

Mathematica [A]  time = 0.0029327, size = 12, normalized size = 1. $\frac{\sec ^2(x)}{2}+\log (\cos (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tan[x]^3,x]

[Out]

Log[Cos[x]] + Sec[x]^2/2

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 17, normalized size = 1.4 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}-{\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)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.924025, size = 27, normalized size = 2.25 \begin{align*} -\frac{1}{2 \,{\left (\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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.07571, size = 57, normalized size = 4.75 \begin{align*} \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)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.086282, size = 12, normalized size = 1. \begin{align*} \log{\left (\cos{\left (x \right )} \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,x)

[Out]

log(cos(x)) + 1/(2*cos(x)**2)

________________________________________________________________________________________

Giac [A]  time = 1.07701, size = 22, normalized size = 1.83 \begin{align*} \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)^3,x, algorithm="giac")

[Out]

1/2*tan(x)^2 - 1/2*log(tan(x)^2 + 1)