### 3.96 $$\int \sec ^2(x) \tan (x) \, dx$$

Optimal. Leaf size=8 $\frac{\sec ^2(x)}{2}$

[Out]

Sec[x]^2/2

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Rubi [A]  time = 0.0116286, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {2606, 30} $\frac{\sec ^2(x)}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sec[x]^2*Tan[x],x]

[Out]

Sec[x]^2/2

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sec ^2(x) \tan (x) \, dx &=\operatorname{Subst}(\int x \, dx,x,\sec (x))\\ &=\frac{\sec ^2(x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0040826, size = 8, normalized size = 1. $\frac{\sec ^2(x)}{2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sec[x]^2*Tan[x],x]

[Out]

Sec[x]^2/2

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Maple [A]  time = 0.01, size = 7, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sec \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2/cot(x),x)

[Out]

1/2*sec(x)^2

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Maxima [A]  time = 0.924727, size = 14, normalized size = 1.75 \begin{align*} -\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(sec(x)^2/cot(x),x, algorithm="maxima")

[Out]

-1/2/(sin(x)^2 - 1)

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Fricas [A]  time = 2.04025, size = 19, normalized size = 2.38 \begin{align*} \frac{1}{2 \, \cos \left (x\right )^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/cot(x),x, algorithm="fricas")

[Out]

1/2/cos(x)^2

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Sympy [A]  time = 0.067394, size = 7, normalized size = 0.88 \begin{align*} \frac{1}{2 \cos ^{2}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2/cot(x),x)

[Out]

1/(2*cos(x)**2)

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Giac [A]  time = 1.05376, size = 8, normalized size = 1. \begin{align*} \frac{1}{2 \, \cos \left (x\right )^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/cot(x),x, algorithm="giac")

[Out]

1/2/cos(x)^2