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

Optimal. Leaf size=15 \[ -\frac {e^x \sin (x)}{1-\cos (x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2326} \begin {gather*} -\frac {e^x \sin (x)}{1-\cos (x)} \end {gather*}

Antiderivative was successfully verified.

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

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 11, normalized size = 0.73 \begin {gather*} -e^x \cot \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.07, size = 21, normalized size = 1.40

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 4.40, size = 22, normalized size = 1.47 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 1.31, size = 12, normalized size = 0.80 \begin {gather*} -\frac {{\left (\cos \left (x\right ) + 1\right )} e^{x}}{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(x) + 1)*e^x/sin(x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\sin {\left (x \right )} - 1\right ) e^{x}}{\cos {\left (x \right )} - 1}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.64, size = 10, normalized size = 0.67 \begin {gather*} -\frac {e^{x}}{\tan \left (\frac {1}{2} \, x\right )} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.39, size = 8, normalized size = 0.53 \begin {gather*} -\mathrm {cot}\left (\frac {x}{2}\right )\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________