3.3.48 \(\int \frac {1}{2 \sin (x)+\sin (2 x)} \, dx\) [248]

Optimal. Leaf size=24 \[ \frac {1}{4} \log \left (\tan \left (\frac {x}{2}\right )\right )+\frac {1}{8} \tan ^2\left (\frac {x}{2}\right ) \]

[Out]

1/4*ln(tan(1/2*x))+1/8*tan(1/2*x)^2

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 14} \begin {gather*} \frac {1}{8} \tan ^2\left (\frac {x}{2}\right )+\frac {1}{4} \log \left (\tan \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*Sin[x] + Sin[2*x])^(-1),x]

[Out]

Log[Tan[x/2]]/4 + Tan[x/2]^2/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \frac {1}{2 \sin (x)+\sin (2 x)} \, dx &=2 \text {Subst}\left (\int \frac {1+x^2}{8 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{4} \log \left (\tan \left (\frac {x}{2}\right )\right )+\frac {1}{8} \tan ^2\left (\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 39, normalized size = 1.62 \begin {gather*} \frac {1-2 \cos ^2\left (\frac {x}{2}\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )}{4 (1+\cos (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*Sin[x] + Sin[2*x])^(-1),x]

[Out]

(1 - 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(4*(1 + Cos[x]))

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Maple [A]
time = 0.09, size = 24, normalized size = 1.00

method	result	size
default	\(\frac {1}{4 \cos \left (x \right )+4}-\frac {\ln \left (1+\cos \left (x \right )\right )}{8}+\frac {\ln \left (\cos \left (x \right )-1\right )}{8}\)	\(24\)
risch	\(\frac {{\mathrm e}^{i x}}{2 \left (1+{\mathrm e}^{i x}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{4}-\frac {\ln \left (1+{\mathrm e}^{i x}\right )}{4}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*sin(x)+sin(2*x)),x,method=_RETURNVERBOSE)

[Out]

1/4/(1+cos(x))-1/8*ln(1+cos(x))+1/8*ln(cos(x)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (16) = 32\).
time = 1.82, size = 220, normalized size = 9.17 \begin {gather*} \frac {4 \, \cos \left (2 \, x\right ) \cos \left (x\right ) + 8 \, \cos \left (x\right )^{2} - {\left (2 \, {\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + {\left (2 \, {\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 8 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right )}{8 \, {\left (2 \, {\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*sin(x)+sin(2*x)),x, algorithm="maxima")

[Out]

1/8*(4*cos(2*x)*cos(x) + 8*cos(x)^2 - (2*(2*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 4*cos(x)^2 + sin(2*x)^2 + 4*si
n(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + (2*(2*cos(x) + 1)*cos(2*x
) + cos(2*x)^2 + 4*cos(x)^2 + sin(2*x)^2 + 4*sin(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1)*log(cos(x)^2 + sin(x
)^2 - 2*cos(x) + 1) + 4*sin(2*x)*sin(x) + 8*sin(x)^2 + 4*cos(x))/(2*(2*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 4*c
os(x)^2 + sin(2*x)^2 + 4*sin(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).
time = 0.71, size = 35, normalized size = 1.46 \begin {gather*} -\frac {{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - {\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2}{8 \, {\left (\cos \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*sin(x)+sin(2*x)),x, algorithm="fricas")

[Out]

-1/8*((cos(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + 1)*log(-1/2*cos(x) + 1/2) - 2)/(cos(x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{2 \sin {\left (x \right )} + \sin {\left (2 x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*sin(x)+sin(2*x)),x)

[Out]

Integral(1/(2*sin(x) + sin(2*x)), x)

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Giac [A]
time = 0.46, size = 28, normalized size = 1.17 \begin {gather*} -\frac {\cos \left (x\right ) - 1}{8 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {1}{8} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*sin(x)+sin(2*x)),x, algorithm="giac")

[Out]

-1/8*(cos(x) - 1)/(cos(x) + 1) + 1/8*log(-(cos(x) - 1)/(cos(x) + 1))

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Mupad [B]
time = 0.34, size = 16, normalized size = 0.67 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{4}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(2*x) + 2*sin(x)),x)

[Out]

log(tan(x/2))/4 + tan(x/2)^2/8

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