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

Optimal. Leaf size=13 \[ -\frac{e^x \cos (x)}{\sin (x)+1} \]

[Out]

-((E^x*Cos[x])/(1 + Sin[x]))

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Rubi [A]  time = 0.0230758, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2288} \[ -\frac{e^x \cos (x)}{\sin (x)+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^x*Cos[x])/(1 + Sin[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-\cos (x))}{1+\sin (x)} \, dx &=-\frac{e^x \cos (x)}{1+\sin (x)}\\ \end{align*}

Mathematica [A]  time = 0.0630134, size = 23, normalized size = 1.77 \[ -\frac{e^x \left (\cot \left (\frac{x}{2}\right )-1\right )}{\cot \left (\frac{x}{2}\right )+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^x*(-1 + Cot[x/2]))/(1 + Cot[x/2]))

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Maple [B]  time = 0.056, size = 51, normalized size = 3.9 \begin{align*}{ \left ({{\rm e}^{x}}\tan \left ({\frac{x}{2}} \right ) +{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-{{\rm e}^{x}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1} \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(x)*tan(1/2*x)+exp(x)*tan(1/2*x)^3-exp(x)*tan(1/2*x)^2-exp(x))/(tan(1/2*x)^2+1)/(1+tan(1/2*x))

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Maxima [A]  time = 1.21953, size = 30, normalized size = 2.31 \begin{align*} -\frac{2 \, \cos \left (x\right ) e^{x}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.85127, size = 76, normalized size = 5.85 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} e^{x} - e^{x} \sin \left (x\right )}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10581, size = 28, normalized size = 2.15 \begin{align*} \frac{e^{x} \tan \left (\frac{1}{2} \, x\right ) - e^{x}}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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