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

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

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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} \[ \frac {\tan (x)}{2} \]

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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc (2 x) \tan (x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.16, size = 4, normalized size = 0.67 \[ \frac {1}{2} \, \tan \relax (x) \]

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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giac [A]  time = 1.01, size = 4, normalized size = 0.67 \[ \frac {1}{2} \, \tan \relax (x) \]

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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maple [A]  time = 0.05, size = 5, normalized size = 0.83

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

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]  time = 0.43, size = 27, normalized size = 4.50 \[ \frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1} \]

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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mupad [B]  time = 0.18, size = 4, normalized size = 0.67 \[ \frac {\mathrm {tan}\relax (x)}{2} \]

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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sympy [B]  time = 0.79, size = 7, normalized size = 1.17 \[ \frac {\sin {\relax (x )}}{2 \cos {\relax (x )}} \]

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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