\(\int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx\) [563]

Optimal result

Integrand size = 16, antiderivative size = 13 \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=-\frac {e^x \cos (x)}{1+\sin (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2326} \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=-\frac {e^x \cos (x)}{\sin (x)+1} \]

[In]

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

[Out]

-((E^x*Cos[x])/(1 + Sin[x]))

Rule 2326

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*} \text {integral}& = -\frac {e^x \cos (x)}{1+\sin (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=-\frac {e^x \cos (x)}{1+\sin (x)} \]

[In]

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

[Out]

-((E^x*Cos[x])/(1 + Sin[x]))

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{x} \left (\tan \left (\frac {x}{2}\right )-1\right )}{1+\tan \left (\frac {x}{2}\right )}\)	\(18\)
risch	\(-i {\mathrm e}^{x}-\frac {2 \,{\mathrm e}^{x}}{i+{\mathrm e}^{i x}}\)	\(21\)
norman	\(\frac {{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )+{\mathrm e}^{x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=-\frac {{\left (\cos \left (x\right ) + 1\right )} e^{x} - e^{x} \sin \left (x\right )}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \]

[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)

Sympy [F]

\[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=- \int \left (- \frac {e^{x}}{\sin {\left (x \right )} + 1}\right )\, dx - \int \frac {e^{x} \cos {\left (x \right )}}{\sin {\left (x \right )} + 1}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=-\frac {2 \, \cos \left (x\right ) e^{x}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=\frac {e^{x} \tan \left (\frac {1}{2} \, x\right ) - e^{x}}{\tan \left (\frac {1}{2} \, x\right ) + 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx=-{\mathrm {e}}^x\,1{}\mathrm {i}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{x\,1{}\mathrm {i}}+1{}\mathrm {i}} \]

[In]

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

[Out]

- exp(x)*1i - (2*exp(x))/(exp(x*1i) + 1i)