3.6.62 \(\int \frac {e^x (1+\cos (x))}{1+\sin (x)} \, dx\) [562]

Optimal. Leaf size=43 \[ (-2-2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;i e^{i x}\right )+\frac {e^x \cos (x)}{1+\sin (x)} \]

[Out]

(-2-2*I)*exp((1+I)*x)*hypergeom([2, 1-I],[2-I],I*exp(I*x))+exp(x)*cos(x)/(1+sin(x))

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4550, 4547, 4527, 2225, 2283, 2326} \begin {gather*} 4 i e^x \text {Hypergeometric2F1}\left (i,1,1+i,-i e^{-i x}\right )-2 i e^x-\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]

(-2*I)*E^x + (4*I)*E^x*Hypergeometric2F1[I, 1, 1 + I, (-I)/E^(I*x)] - (E^x*Cos[x])/(1 + Sin[x])

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

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]

Rule 4527

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Tan[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Dist[I^n, Int[ExpandIntegran
d[F^(c*(a + b*x))*((1 - E^(2*I*(d + e*x)))^n/(1 + E^(2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e
}, x] && IntegerQ[n]

Rule 4547

Int[Cos[(d_.) + (e_.)*(x_)]^(m_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_) + (g_.)*Sin[(d_.) + (e_.)*(x_)])^(n_
.), x_Symbol] :> Dist[g^n, Int[F^(c*(a + b*x))*Tan[f*(Pi/(4*g)) - d/2 - e*(x/2)]^m, x], x] /; FreeQ[{F, a, b,
c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]

Rule 4550

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_)))*(Cos[(d_.) + (e_.)*(x_)]*(i_.) + (h_)))/((f_) + (g_.)*Sin[(d_.) + (e_.)
*(x_)]), x_Symbol] :> Dist[2*i, Int[F^(c*(a + b*x))*(Cos[d + e*x]/(f + g*Sin[d + e*x])), x], x] + Int[F^(c*(a
+ b*x))*((h - i*Cos[d + e*x])/(f + g*Sin[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2
- g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]

Rubi steps

\begin {align*} \int \frac {e^x (1+\cos (x))}{1+\sin (x)} \, dx &=2 \int \frac {e^x \cos (x)}{1+\sin (x)} \, dx+\int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx\\ &=-\frac {e^x \cos (x)}{1+\sin (x)}+2 \int e^x \tan \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx\\ &=-\frac {e^x \cos (x)}{1+\sin (x)}+2 i \int \left (-e^x+\frac {2 e^x}{1+e^{2 i \left (\frac {\pi }{4}-\frac {x}{2}\right )}}\right ) \, dx\\ &=-\frac {e^x \cos (x)}{1+\sin (x)}-2 i \int e^x \, dx+4 i \int \frac {e^x}{1+e^{2 i \left (\frac {\pi }{4}-\frac {x}{2}\right )}} \, dx\\ &=-2 i e^x+4 i e^x \, _2F_1\left (i,1;1+i;-i e^{-i x}\right )-\frac {e^x \cos (x)}{1+\sin (x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 73, normalized size = 1.70 \begin {gather*} \frac {1}{2} (1+\cos (x)) \sec ^2\left (\frac {x}{2}\right ) \left (-4 i \, _2F_1(-i,1;1-i;i \cos (x)-\sin (x)) (\cosh (x)+\sinh (x))+\frac {e^x \left ((-1+2 i)+(1+2 i) \tan \left (\frac {x}{2}\right )\right )}{1+\tan \left (\frac {x}{2}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + Cos[x])*Sec[x/2]^2*((-4*I)*Hypergeometric2F1[-I, 1, 1 - I, I*Cos[x] - Sin[x]]*(Cosh[x] + Sinh[x]) + (E^x
*((-1 + 2*I) + (1 + 2*I)*Tan[x/2]))/(1 + Tan[x/2])))/2

________________________________________________________________________________________

Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} \left (1+\cos \left (x \right )\right )}{\sin \left (x \right )+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 - 2*(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1)*integrate(cos(x)*e^x/(cos(x)^2 + sin(x)^2 + 2*sin(x) +
1), x))/(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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]

integral((cos(x) + 1)*e^x/(sin(x) + 1), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\cos {\left (x \right )} + 1\right ) e^{x}}{\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((cos(x) + 1)*exp(x)/(sin(x) + 1), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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]

integrate((cos(x) + 1)*e^x/(sin(x) + 1), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (\cos \left (x\right )+1\right )}{\sin \left (x\right )+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________