3.6.63 \(\int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx\) [563]

Optimal. Leaf size=13 \[ -\frac {e^x \cos (x)}{1+\sin (x)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 13, 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)}{\sin (x)+1} \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 = 0.05, size = 23, normalized size = 1.77 \begin {gather*} -\frac {e^x \left (-1+\cot \left (\frac {x}{2}\right )\right )}{1+\cot \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 + Cot[x/2]))/(1 + Cot[x/2]))

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Maple [C] Result contains complex when optimal does not.
time = 0.08, size = 21, normalized size = 1.62

method	result	size
risch	\(-i {\mathrm e}^{x}-\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{i x}+i}\)	\(21\)
norman	\(\frac {{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )+{\mathrm e}^{x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 4.53, size = 22, normalized size = 1.69 \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 = 0.81, size = 24, normalized size = 1.85 \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 \left (- \frac {e^{x}}{\sin {\left (x \right )} + 1}\right )\, 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 = 1.08, size = 21, normalized size = 1.62 \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 = 0.42, size = 20, normalized size = 1.54 \begin {gather*} -{\mathrm {e}}^x\,1{}\mathrm {i}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{x\,1{}\mathrm {i}}+1{}\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)*1i - (2*exp(x))/(exp(x*1i) + 1i)

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