3.8 \(\int \csc (2 x) \tan (x) \, dx\)

Optimal. Leaf size=6 \[ \frac{\tan (x)}{2} \]

[Out]

Tan[x]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0235679, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {8} \[ \frac{\tan (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[2*x]*Tan[x],x]

[Out]

Tan[x]/2

Rule 8

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

Rubi steps

\begin{align*} \int \csc (2 x) \tan (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{2} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0128772, size = 6, normalized size = 1. \[ \frac{\tan (x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*x]*Tan[x],x]

[Out]

Tan[x]/2

________________________________________________________________________________________

Maple [A]  time = 0.014, size = 5, normalized size = 0.8 \begin{align*}{\frac{\tan \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/sin(2*x),x)

[Out]

1/2*tan(x)

________________________________________________________________________________________

Maxima [B]  time = 0.932013, size = 36, normalized size = 6. \begin{align*} \frac{\sin \left (2 \, x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/sin(2*x),x, algorithm="maxima")

[Out]

sin(2*x)/(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.86448, size = 16, normalized size = 2.67 \begin{align*} \frac{1}{2} \, \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/sin(2*x),x, algorithm="fricas")

[Out]

1/2*tan(x)

________________________________________________________________________________________

Sympy [B]  time = 0.716762, size = 7, normalized size = 1.17 \begin{align*} \frac{\sin{\left (x \right )}}{2 \cos{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/sin(2*x),x)

[Out]

sin(x)/(2*cos(x))

________________________________________________________________________________________

Giac [A]  time = 1.05491, size = 5, normalized size = 0.83 \begin{align*} \frac{1}{2} \, \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/sin(2*x),x, algorithm="giac")

[Out]

1/2*tan(x)