3.558 \(\int \frac {e^x (1+\sin (x))}{1+\cos (x)} \, dx\)

Optimal. Leaf size=12 \[ \frac {e^x \sin (x)}{\cos (x)+1} \]

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Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2288} \[ \frac {e^x \sin (x)}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(E^x*(1 + Sin[x]))/(1 + Cos[x]),x]

[Out]

(E^x*Sin[x])/(1 + Cos[x])

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {align*} \int \frac {e^x (1+\sin (x))}{1+\cos (x)} \, dx &=\frac {e^x \sin (x)}{1+\cos (x)}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 10, normalized size = 0.83 \[ e^x \tan \left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(1 + Sin[x]))/(1 + Cos[x]),x]

[Out]

E^x*Tan[x/2]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1+\sin (x))}{1+\cos (x)} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(E^x*(1 + Sin[x]))/(1 + Cos[x]),x]

[Out]

Could not integrate

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fricas [A]  time = 1.30, size = 11, normalized size = 0.92 \[ \frac {e^{x} \sin \relax (x)}{\cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x*sin(x)/(cos(x) + 1)

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giac [A]  time = 0.63, size = 7, normalized size = 0.58 \[ e^{x} \tan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x*tan(1/2*x)

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maple [A]  time = 0.08, size = 8, normalized size = 0.67

method	result	size
norman	\({\mathrm e}^{x} \tan \left (\frac {x}{2}\right )\)	\(8\)
risch	\(-i {\mathrm e}^{x}+\frac {2 i {\mathrm e}^{x}}{{\mathrm e}^{i x}+1}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*tan(1/2*x)

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maxima [A]  time = 0.72, size = 22, normalized size = 1.83 \[ \frac {2 \, e^{x} \sin \relax (x)}{\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.37, size = 7, normalized size = 0.58 \[ \mathrm {tan}\left (\frac {x}{2}\right )\,{\mathrm {e}}^x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(sin(x) + 1))/(cos(x) + 1),x)

[Out]

tan(x/2)*exp(x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sin {\relax (x )} + 1\right ) e^{x}}{\cos {\relax (x )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1+sin(x))/(1+cos(x)),x)

[Out]

Integral((sin(x) + 1)*exp(x)/(cos(x) + 1), x)

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