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

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

[Out]

-((E^x*Sin[x])/(1 - Cos[x]))

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Rubi [A]  time = 0.0299952, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.21783, 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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Maple [B]  time = 0.047, size = 33, normalized size = 2.2 \begin{align*}{ \left ( -{{\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 ( \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-sin(x))/(1-cos(x)),x)

[Out]

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

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Maxima [A]  time = 1.27011, size = 30, normalized size = 2. \begin{align*} -\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{align*}

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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Fricas [A]  time = 2.16965, size = 35, normalized size = 2.33 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} e^{x}}{\sin \left (x\right )} \end{align*}

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

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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Giac [A]  time = 1.1411, size = 14, normalized size = 0.93 \begin{align*} -\frac{e^{x}}{\tan \left (\frac{1}{2} \, x\right )} \end{align*}

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)