\(\int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx\) [556]

Optimal result

Integrand size = 18, antiderivative size = 15 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=-\frac {e^x \sin (x)}{1-\cos (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2326} \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=-\frac {e^x \sin (x)}{1-\cos (x)} \]

[In]

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

[Out]

-((E^x*Sin[x])/(1 - Cos[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 \sin (x)}{1-\cos (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=\frac {e^x \sin (x)}{-1+\cos (x)} \]

[In]

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

[Out]

(E^x*Sin[x])/(-1 + Cos[x])

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73

method	result	size
parallelrisch	\(-\frac {{\mathrm e}^{x}}{\tan \left (\frac {x}{2}\right )}\)	\(11\)
risch	\(-i {\mathrm e}^{x}-\frac {2 i {\mathrm e}^{x}}{{\mathrm e}^{i x}-1}\)	\(21\)
norman	\(\frac {-{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=-\frac {{\left (\cos \left (x\right ) + 1\right )} e^{x}}{\sin \left (x\right )} \]

[In]

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

[Out]

-(cos(x) + 1)*e^x/sin(x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=-\frac {2 \, e^{x} \sin \left (x\right )}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=-\frac {e^{x}}{\tan \left (\frac {1}{2} \, x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx=-\mathrm {cot}\left (\frac {x}{2}\right )\,{\mathrm {e}}^x \]

[In]

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

[Out]

-cot(x/2)*exp(x)