\(\int \csc (2 x) \tan (x) \, dx\) [8]

Optimal result

Integrand size = 7, antiderivative size = 6 \[ \int \csc (2 x) \tan (x) \, dx=\frac {\tan (x)}{2} \]

[Out]

1/2*tan(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {8} \[ \int \csc (2 x) \tan (x) \, dx=\frac {\tan (x)}{2} \]

[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*} \text {integral}& = \text {Subst}\left (\int \frac {1}{2} \, dx,x,\tan (x)\right ) \\ & = \frac {\tan (x)}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \csc (2 x) \tan (x) \, dx=\frac {\tan (x)}{2} \]

[In]

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

[Out]

Tan[x]/2

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\tan \left (x \right )}{2}\)	\(5\)
default	\(\frac {\tan \left (x \right )}{2}\)	\(5\)
norman	\(\frac {\tan \left (x \right )}{2}\)	\(5\)
parallelrisch	\(\frac {\tan \left (x \right )}{2}\)	\(5\)
risch	\(\frac {i}{{\mathrm e}^{2 i x}+1}\)	\(13\)

[In]

int(tan(x)/sin(2*x),x,method=_RETURNVERBOSE)

[Out]

1/2*tan(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.67 \[ \int \csc (2 x) \tan (x) \, dx=\frac {1}{2} \, \tan \left (x\right ) \]

[In]

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

[Out]

1/2*tan(x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7 vs. \(2 (3) = 6\).

Time = 0.32 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \csc (2 x) \tan (x) \, dx=\frac {\sin {\left (x \right )}}{2 \cos {\left (x \right )}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (4) = 8\).

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 4.50 \[ \int \csc (2 x) \tan (x) \, dx=\frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.67 \[ \int \csc (2 x) \tan (x) \, dx=\frac {1}{2} \, \tan \left (x\right ) \]

[In]

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

[Out]

1/2*tan(x)

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.67 \[ \int \csc (2 x) \tan (x) \, dx=\frac {\mathrm {tan}\left (x\right )}{2} \]

[In]

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

[Out]

tan(x)/2