3.574 \(\int \frac{(d+e e^{h+i x}) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx\)

Optimal. Leaf size=428 \[ -\frac{g \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{g \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{2 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{2 g \left (b-\sqrt{b^2-4 a c}\right )} \]

[Out]

((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^2)/(2*(b + Sqrt[b^2 - 4*a*c])*g) + ((e + (2*c*d - b*e)/Sqrt[b
^2 - 4*a*c])*(f + g*x)^2)/(2*(b - Sqrt[b^2 - 4*a*c])*g) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)*Log
[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i) - ((e - (2*c*d - b*e)/Sqrt[b^2 -
4*a*c])*(f + g*x)*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i) - ((e + (2*c
*d - b*e)/Sqrt[b^2 - 4*a*c])*g*PolyLog[2, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c]
)*i^2) - ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*PolyLog[2, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b
+ Sqrt[b^2 - 4*a*c])*i^2)

________________________________________________________________________________________

Rubi [A]  time = 0.582517, antiderivative size = 428, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2265, 2184, 2190, 2279, 2391} \[ -\frac{g \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{g \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{2 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{2 g \left (b-\sqrt{b^2-4 a c}\right )} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*E^(h + i*x))*(f + g*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)),x]

[Out]

((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^2)/(2*(b + Sqrt[b^2 - 4*a*c])*g) + ((e + (2*c*d - b*e)/Sqrt[b
^2 - 4*a*c])*(f + g*x)^2)/(2*(b - Sqrt[b^2 - 4*a*c])*g) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)*Log
[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i) - ((e - (2*c*d - b*e)/Sqrt[b^2 -
4*a*c])*(f + g*x)*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i) - ((e + (2*c
*d - b*e)/Sqrt[b^2 - 4*a*c])*g*PolyLog[2, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c]
)*i^2) - ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*PolyLog[2, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b
+ Sqrt[b^2 - 4*a*c])*i^2)

Rule 2265

Int[(((i_.)*(F_)^(u_) + (h_))*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Simplify[(2*c*h - b*i)/q] + i, Int[(f + g*x)^m/(b - q + 2*c*F^u), x]
, x] - Dist[Simplify[(2*c*h - b*i)/q] - i, Int[(f + g*x)^m/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f,
g, h, i}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d+e e^{h+574 x}\right ) (f+g x)}{a+b e^{h+574 x}+c e^{2 h+1148 x}} \, dx &=-\left (\left (-e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{f+g x}{b+\sqrt{b^2-4 a c}+2 c e^{h+574 x}} \, dx\right )+\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{f+g x}{b-\sqrt{b^2-4 a c}+2 c e^{h+574 x}} \, dx\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (2 c \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{e^{h+574 x} (f+g x)}{b+\sqrt{b^2-4 a c}+2 c e^{h+574 x}} \, dx}{b+\sqrt{b^2-4 a c}}-\frac{\left (2 c \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{e^{h+574 x} (f+g x)}{b-\sqrt{b^2-4 a c}+2 c e^{h+574 x}} \, dx}{b-\sqrt{b^2-4 a c}}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac{2 c e^{h+574 x}}{b-\sqrt{b^2-4 a c}}\right )}{574 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac{2 c e^{h+574 x}}{b+\sqrt{b^2-4 a c}}\right )}{574 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \int \log \left (1+\frac{2 c e^{h+574 x}}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{574 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \int \log \left (1+\frac{2 c e^{h+574 x}}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{574 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac{2 c e^{h+574 x}}{b-\sqrt{b^2-4 a c}}\right )}{574 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac{2 c e^{h+574 x}}{b+\sqrt{b^2-4 a c}}\right )}{574 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^{h+574 x}\right )}{329476 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^{h+574 x}\right )}{329476 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac{2 c e^{h+574 x}}{b-\sqrt{b^2-4 a c}}\right )}{574 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac{2 c e^{h+574 x}}{b+\sqrt{b^2-4 a c}}\right )}{574 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g \text{Li}_2\left (-\frac{2 c e^{h+574 x}}{b-\sqrt{b^2-4 a c}}\right )}{329476 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g \text{Li}_2\left (-\frac{2 c e^{h+574 x}}{b+\sqrt{b^2-4 a c}}\right )}{329476 \left (b+\sqrt{b^2-4 a c}\right )}\\ \end{align*}

Mathematica [A]  time = 1.76181, size = 677, normalized size = 1.58 \[ -\frac{g \left (d \sqrt{-\left (b^2-4 a c\right )^2}+b d \sqrt{4 a c-b^2}-2 a e \sqrt{4 a c-b^2}\right ) \text{PolyLog}\left (2,\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}-b}\right )+g \left (d \sqrt{-\left (b^2-4 a c\right )^2}-b d \sqrt{4 a c-b^2}+2 a e \sqrt{4 a c-b^2}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )+i \left (d f \sqrt{-\left (b^2-4 a c\right )^2} \log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right )+2 b d f \sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{b+2 c e^{h+i x}}{\sqrt{4 a c-b^2}}\right )-2 d f i x \sqrt{-\left (b^2-4 a c\right )^2}+d g x \sqrt{-\left (b^2-4 a c\right )^2} \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )+b d g x \sqrt{4 a c-b^2} \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )+d g x \sqrt{-\left (b^2-4 a c\right )^2} \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )-b d g x \sqrt{4 a c-b^2} \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )+d g i x^2 \left (-\sqrt{-\left (b^2-4 a c\right )^2}\right )+4 a e f \sqrt{4 a c-b^2} \tanh ^{-1}\left (\frac{b+2 c e^{h+i x}}{\sqrt{b^2-4 a c}}\right )-2 a e g x \sqrt{4 a c-b^2} \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )+2 a e g x \sqrt{4 a c-b^2} \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )\right )}{2 a i^2 \sqrt{-\left (b^2-4 a c\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*E^(h + i*x))*(f + g*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)),x]

[Out]

-(i*(-2*Sqrt[-(b^2 - 4*a*c)^2]*d*f*i*x - Sqrt[-(b^2 - 4*a*c)^2]*d*g*i*x^2 + 2*b*Sqrt[b^2 - 4*a*c]*d*f*ArcTan[(
b + 2*c*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] + 4*a*Sqrt[-b^2 + 4*a*c]*e*f*ArcTanh[(b + 2*c*E^(h + i*x))/Sqrt[b^2 -
4*a*c]] + Sqrt[-(b^2 - 4*a*c)^2]*d*g*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + b*Sqrt[-b^2 + 4*a
*c]*d*g*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 2*a*Sqrt[-b^2 + 4*a*c]*e*g*x*Log[1 + (2*c*E^(h
+ i*x))/(b - Sqrt[b^2 - 4*a*c])] + Sqrt[-(b^2 - 4*a*c)^2]*d*g*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*
c])] - b*Sqrt[-b^2 + 4*a*c]*d*g*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 2*a*Sqrt[-b^2 + 4*a*c]*
e*g*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + Sqrt[-(b^2 - 4*a*c)^2]*d*f*Log[a + E^(h + i*x)*(b +
c*E^(h + i*x))]) + (Sqrt[-(b^2 - 4*a*c)^2]*d + b*Sqrt[-b^2 + 4*a*c]*d - 2*a*Sqrt[-b^2 + 4*a*c]*e)*g*PolyLog[2
, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] + (Sqrt[-(b^2 - 4*a*c)^2]*d - b*Sqrt[-b^2 + 4*a*c]*d + 2*a*Sqrt[
-b^2 + 4*a*c]*e)*g*PolyLog[2, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/(2*a*Sqrt[-(b^2 - 4*a*c)^2]*i^2)

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Maple [B]  time = 0.032, size = 1261, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*exp(i*x+h))*(g*x+f)/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)

[Out]

d*f/i/a*ln(exp(i*x))-1/2*d*f/i/a*ln(a+b*exp(i*x)*exp(h)+c*exp(i*x)^2*exp(2*h))-d*f/i/a*exp(h)*b/(4*a*c*exp(2*h
)-exp(h)^2*b^2)^(1/2)*arctan((exp(h)*b+2*exp(2*h)*exp(i*x)*c)/(4*a*c*exp(2*h)-exp(h)^2*b^2)^(1/2))+1/2*d*g/a*x
^2-1/2*d*g/i/a*x/(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)*exp(h)*ln((2*exp(2*h)*exp(i*x)*c+exp(h)*b-(exp(h)^2*b^2-4
*a*c*exp(2*h))^(1/2))/(exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))*b+1/2*d*g/i/a*x/(exp(h)^2*b^2-4*a*c*exp(
2*h))^(1/2)*exp(h)*ln((2*exp(2*h)*exp(i*x)*c+exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b+(exp(h)^2
*b^2-4*a*c*exp(2*h))^(1/2)))*b-1/2*d*g/i/a*x*ln((2*exp(2*h)*exp(i*x)*c+exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^
(1/2))/(exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))-1/2*d*g/i/a*x*ln((2*exp(2*h)*exp(i*x)*c+exp(h)*b+(exp(h
)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))+1/2*d*g/i^2/a/(exp(h)^2*b^2-4*a
*c*exp(2*h))^(1/2)*exp(h)*dilog((2*exp(2*h)*exp(i*x)*c+exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b
+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))*b-1/2*d*g/i^2/a/(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)*exp(h)*dilog((2*exp
(2*h)*exp(i*x)*c+exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))
*b-1/2*d*g/i^2/a*dilog((2*exp(2*h)*exp(i*x)*c+exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b+(exp(h)^
2*b^2-4*a*c*exp(2*h))^(1/2)))-1/2*d*g/i^2/a*dilog((2*exp(2*h)*exp(i*x)*c+exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h)
)^(1/2))/(exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))+2*e*exp(h)*f/i/(4*a*c*exp(2*h)-exp(h)^2*b^2)^(1/2)*ar
ctan((exp(h)*b+2*exp(2*h)*exp(i*x)*c)/(4*a*c*exp(2*h)-exp(h)^2*b^2)^(1/2))+e*exp(h)*g/i*x/(exp(h)^2*b^2-4*a*c*
exp(2*h))^(1/2)*ln((2*exp(2*h)*exp(i*x)*c+exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b-(exp(h)^2*b^
2-4*a*c*exp(2*h))^(1/2)))-e*exp(h)*g/i*x/(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)*ln((2*exp(2*h)*exp(i*x)*c+exp(h)*
b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))+e*exp(h)*g/i^2/(exp(h)^
2*b^2-4*a*c*exp(2*h))^(1/2)*dilog((2*exp(2*h)*exp(i*x)*c+exp(h)*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)
*b-(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))-e*exp(h)*g/i^2/(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)*dilog((2*exp(2*h)*
exp(i*x)*c+exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2))/(exp(h)*b+(exp(h)^2*b^2-4*a*c*exp(2*h))^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34788, size = 1513, normalized size = 3.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="fricas")

[Out]

1/2*((b^2 - 4*a*c)*d*g*i^2*x^2 + 2*(b^2 - 4*a*c)*d*f*i^2*x - ((b^2 - 4*a*c)*d*g + (a*b*d - 2*a^2*e)*g*sqrt((b^
2 - 4*a*c)/a^2))*dilog(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*x + h) + 2*a)/a + 1) - ((b^2 - 4*a
*c)*d*g - (a*b*d - 2*a^2*e)*g*sqrt((b^2 - 4*a*c)/a^2))*dilog(1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) - b*e^
(i*x + h) - 2*a)/a + 1) + ((b^2 - 4*a*c)*d*g*h - (b^2 - 4*a*c)*d*f*i - ((a*b*d - 2*a^2*e)*g*h - (a*b*d - 2*a^2
*e)*f*i)*sqrt((b^2 - 4*a*c)/a^2))*log(2*c*e^(i*x + h) + a*sqrt((b^2 - 4*a*c)/a^2) + b) + ((b^2 - 4*a*c)*d*g*h
- (b^2 - 4*a*c)*d*f*i + ((a*b*d - 2*a^2*e)*g*h - (a*b*d - 2*a^2*e)*f*i)*sqrt((b^2 - 4*a*c)/a^2))*log(2*c*e^(i*
x + h) - a*sqrt((b^2 - 4*a*c)/a^2) + b) - ((b^2 - 4*a*c)*d*g*i*x + (b^2 - 4*a*c)*d*g*h + ((a*b*d - 2*a^2*e)*g*
i*x + (a*b*d - 2*a^2*e)*g*h)*sqrt((b^2 - 4*a*c)/a^2))*log(1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*
x + h) + 2*a)/a) - ((b^2 - 4*a*c)*d*g*i*x + (b^2 - 4*a*c)*d*g*h - ((a*b*d - 2*a^2*e)*g*i*x + (a*b*d - 2*a^2*e)
*g*h)*sqrt((b^2 - 4*a*c)/a^2))*log(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) - b*e^(i*x + h) - 2*a)/a))/((a*
b^2 - 4*a^2*c)*i^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)

[Out]

Integral((d + e*exp(h)*exp(i*x))*(f + g*x)/(a + b*exp(h)*exp(i*x) + c*exp(2*h)*exp(2*i*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}{\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="giac")

[Out]

integrate((g*x + f)*(e*e^(i*x + h) + d)/(c*e^(2*i*x + 2*h) + b*e^(i*x + h) + a), x)