3.3.93 \(\int \frac {\sin (x)}{2 e^x+\cos (x)+\sin (x)} \, dx\) [293]

Optimal. Leaf size=19 \[ \frac {1}{2} \left (x-\log \left (2 e^x+\cos (x)+\sin (x)\right )\right ) \]

[Out]

1/2*x-1/2*ln(2*exp(x)+cos(x)+sin(x))

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Rubi [F]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sin (x)}{2 e^x+\cos (x)+\sin (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sin[x]/(2*E^x + Cos[x] + Sin[x]),x]

[Out]

Defer[Int][Sin[x]/(2*E^x + Cos[x] + Sin[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \frac {\sin (x)}{2 e^x+\cos (x)+\sin (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 21, normalized size = 1.11 \begin {gather*} \frac {x}{2}-\frac {1}{2} \log \left (2 e^x+\cos (x)+\sin (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(2*E^x + Cos[x] + Sin[x]),x]

[Out]

x/2 - Log[2*E^x + Cos[x] + Sin[x]]/2

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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 30, normalized size = 1.58

method	result	size
risch	\(\frac {x}{2}+\frac {i x}{2}-\frac {\ln \left ({\mathrm e}^{2 i x}+\left (2+2 i\right ) {\mathrm e}^{\left (1+i\right ) x}+i\right )}{2}\)	\(30\)
parallelrisch	\(\ln \left (\frac {\sqrt {2}}{\sqrt {\frac {2 \,{\mathrm e}^{x}+\cos \left (x \right )+\sin \left (x \right )}{1+\cos \left (x \right )}}}\right )+\ln \left (\sqrt {\frac {1}{1+\cos \left (x \right )}}\right )+\frac {x}{2}\)	\(37\)
norman	\(\frac {\frac {x}{2}+\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}}{1+\tan ^{2}\left (\frac {x}{2}\right )}+\frac {\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \,{\mathrm e}^{x}+2 \tan \left (\frac {x}{2}\right )+1\right )}{2}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x+1/2*I*x-1/2*ln(exp(2*I*x)+(2+2*I)*exp((1+I)*x)+I)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (16) = 32\).
time = 0.34, size = 91, normalized size = 4.79 \begin {gather*} \frac {1}{2} \, x - \frac {1}{4} \, \log \left (8 \, \cos \left (x\right )^{2} e^{\left (2 \, x\right )} + 8 \, e^{\left (2 \, x\right )} \sin \left (x\right )^{2} + 4 \, {\left (\cos \left (x\right ) e^{x} - e^{x} \sin \left (x\right )\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right ) e^{x} + 2 \, {\left (2 \, \cos \left (x\right ) e^{x} + 2 \, e^{x} \sin \left (x\right ) + 1\right )} \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} + 4 \, e^{x} \sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/4*log(8*cos(x)^2*e^(2*x) + 8*e^(2*x)*sin(x)^2 + 4*(cos(x)*e^x - e^x*sin(x))*cos(2*x) + cos(2*x)^2 +
4*cos(x)*e^x + 2*(2*cos(x)*e^x + 2*e^x*sin(x) + 1)*sin(2*x) + sin(2*x)^2 + 4*e^x*sin(x) + 1)

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Fricas [A]
time = 0.61, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, x - \frac {1}{2} \, \log \left (\cos \left (x\right ) + 2 \, e^{x} + \sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/2*log(cos(x) + 2*e^x + sin(x))

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Sympy [A]
time = 0.07, size = 17, normalized size = 0.89 \begin {gather*} \frac {x}{2} - \frac {\log {\left (2 e^{x} + \sin {\left (x \right )} + \cos {\left (x \right )} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(2*exp(x)+cos(x)+sin(x)),x)

[Out]

x/2 - log(2*exp(x) + sin(x) + cos(x))/2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (16) = 32\).
time = 0.54, size = 119, normalized size = 6.26 \begin {gather*} \frac {1}{2} \, x - \frac {1}{4} \, \log \left (\frac {2 \, {\left (4 \, e^{\left (2 \, x\right )} \tan \left (\frac {1}{2} \, x\right )^{4} - 4 \, e^{x} \tan \left (\frac {1}{2} \, x\right )^{4} + 8 \, e^{x} \tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{4} + 8 \, e^{\left (2 \, x\right )} \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 8 \, e^{x} \tan \left (\frac {1}{2} \, x\right ) + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, e^{\left (2 \, x\right )} + 4 \, e^{x} + 4 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/4*log(2*(4*e^(2*x)*tan(1/2*x)^4 - 4*e^x*tan(1/2*x)^4 + 8*e^x*tan(1/2*x)^3 + tan(1/2*x)^4 + 8*e^(2*x)
*tan(1/2*x)^2 - 4*tan(1/2*x)^3 + 8*e^x*tan(1/2*x) + 2*tan(1/2*x)^2 + 4*e^(2*x) + 4*e^x + 4*tan(1/2*x) + 1)/(ta
n(1/2*x)^4 + 2*tan(1/2*x)^2 + 1))

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Mupad [B]
time = 0.16, size = 22, normalized size = 1.16 \begin {gather*} \frac {x}{2}-\frac {\ln \left (2\,{\mathrm {e}}^x+\sqrt {2}\,\cos \left (x-\frac {\pi }{4}\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x) + 2*exp(x) + sin(x)),x)

[Out]

x/2 - log(2*exp(x) + 2^(1/2)*cos(x - pi/4))/2

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Chatgpt [F] Failed to verify
time = 1.00, size = 21, normalized size = 1.11 \begin {gather*} 2 \arctan \left ({\mathrm e}^{x}+\sin \left (x \right )\right )-2 \ln \left ({\mathrm e}^{x}-\cos \left (x \right )+\sin \left (x \right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(sin(x)/(2*exp(x)+cos(x)+sin(x)),x)

[Out]

2*arctan(exp(x)+sin(x))-2*ln(exp(x)-cos(x)+sin(x))

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