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

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

[Out]

1/2*tan(x)

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Rubi [A]
time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\tan (x)}{2} \end {gather*}

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

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Mathematica [A]
time = 0.01, size = 6, normalized size = 1.00 \begin {gather*} \frac {\tan (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x]/2

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Maple [A]
time = 0.03, size = 5, normalized size = 0.83

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*tan(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (4) = 8\).
time = 1.89, size = 27, normalized size = 4.50 \begin {gather*} \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 {gather*}

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)

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Fricas [A]
time = 1.17, size = 4, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \tan \left (x\right ) \end {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 7 vs. \(2 (3) = 6\).
time = 0.40, size = 7, normalized size = 1.17 \begin {gather*} \frac {\sin {\left (x \right )}}{2 \cos {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.82, size = 4, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \tan \left (x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.18, size = 4, normalized size = 0.67 \begin {gather*} \frac {\mathrm {tan}\left (x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)/2

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