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

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

[Out]

(E^x*Sin[x])/(1 + Cos[x])

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Rubi [A]  time = 0.0277275, antiderivative size = 12, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.180259, 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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Maple [A]  time = 0.034, size = 8, normalized size = 0.7 \begin{align*}{{\rm e}^{x}}\tan \left ({\frac{x}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20976, size = 30, normalized size = 2.5 \begin{align*} \frac{2 \, e^{x} \sin \left (x\right )}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1} \end{align*}

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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Fricas [A]  time = 1.95081, size = 34, normalized size = 2.83 \begin{align*} \frac{e^{x} \sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\sin{\left (x \right )} + 1\right ) e^{x}}{\cos{\left (x \right )} + 1}\, dx \end{align*}

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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Giac [A]  time = 1.15052, size = 9, normalized size = 0.75 \begin{align*} e^{x} \tan \left (\frac{1}{2} \, x\right ) \end{align*}

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)