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

Optimal. Leaf size=14 \[ \frac{e^x \cos (x)}{1-\sin (x)} \]

[Out]

(E^x*Cos[x])/(1 - Sin[x])

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Rubi [A]  time = 0.0240413, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2288} \[ \frac{e^x \cos (x)}{1-\sin (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^x*Cos[x])/(1 - Sin[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+\cos (x))}{1-\sin (x)} \, dx &=\frac{e^x \cos (x)}{1-\sin (x)}\\ \end{align*}

Mathematica [A]  time = 0.0748232, size = 23, normalized size = 1.64 \[ -\frac{e^x \left (\tan \left (\frac{x}{2}\right )+1\right )}{\tan \left (\frac{x}{2}\right )-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^x*(1 + Tan[x/2]))/(-1 + Tan[x/2]))

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Maple [B]  time = 0.069, size = 53, normalized size = 3.8 \begin{align*}{ \left ( -{{\rm e}^{x}}\tan \left ({\frac{x}{2}} \right ) -{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-{{\rm e}^{x}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-exp(x)*tan(1/2*x)-exp(x)*tan(1/2*x)^2-exp(x)*tan(1/2*x)^3-exp(x))/(tan(1/2*x)^2+1)/(tan(1/2*x)-1)

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Maxima [A]  time = 1.21475, size = 30, normalized size = 2.14 \begin{align*} \frac{2 \, \cos \left (x\right ) e^{x}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.81892, size = 74, normalized size = 5.29 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.14211, size = 27, normalized size = 1.93 \begin{align*} -\frac{e^{x} \tan \left (\frac{1}{2} \, x\right ) + e^{x}}{\tan \left (\frac{1}{2} \, x\right ) - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)