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

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

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Rubi [A]  time = 0.03, 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 = {2288} \[ -\frac {e^x \sin (x)}{1-\cos (x)} \]

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*}

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Mathematica [A]  time = 0.23, size = 11, normalized size = 0.73 \[ -e^x \cot \left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.15, size = 12, normalized size = 0.80 \[ -\frac {{\left (\cos \relax (x) + 1\right )} e^{x}}{\sin \relax (x)} \]

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)

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

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)

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maple [C]  time = 0.10, 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)

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maxima [A]  time = 0.72, size = 22, normalized size = 1.47 \[ -\frac {2 \, e^{x} \sin \relax (x)}{\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1} \]

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

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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sin {\relax (x )} - 1\right ) e^{x}}{\cos {\relax (x )} - 1}\, dx \]

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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