3.6.61 \(\int \frac {e^x (1+\cos (x))}{1-\sin (x)} \, dx\) [561]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {2326} \begin {gather*} \frac {e^x \cos (x)}{1-\sin (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.06, size = 23, normalized size = 1.64 \begin {gather*} -\frac {e^x \left (1+\tan \left (\frac {x}{2}\right )\right )}{-1+\tan \left (\frac {x}{2}\right )} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.10, size = 21, normalized size = 1.50

method	result	size
risch	\(-i {\mathrm e}^{x}+\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{i x}-i}\)	\(21\)
norman	\(\frac {-{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )-{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right )}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 3.77, size = 22, normalized size = 1.57 \begin {gather*} \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 {gather*}

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 = 1.09, size = 24, normalized size = 1.71 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 {gather*}

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 = 0.82, size = 20, normalized size = 1.43 \begin {gather*} -\frac {e^{x} \tan \left (\frac {1}{2} \, x\right ) + e^{x}}{\tan \left (\frac {1}{2} \, x\right ) - 1} \end {gather*}

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)

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Mupad [B]
time = 0.37, size = 24, normalized size = 1.71 \begin {gather*} -\frac {{\mathrm {e}}^x\,\left (-1+{\mathrm {e}}^{x\,1{}\mathrm {i}}\,1{}\mathrm {i}\right )}{{\mathrm {e}}^{x\,1{}\mathrm {i}}-\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(x)*(exp(x*1i)*1i - 1))/(exp(x*1i) - 1i)

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