\(\int \frac {(d+e e^{h+i x}) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx\) [574]

Optimal result

Integrand size = 42, antiderivative size = 428 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2}{2 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x) \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2} \]

[Out]

1/2*(g*x+f)^2*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/g/(b-(-4*a*c+b^2)^(1/2))-(g*x+f)*ln(1+2*c*exp(i*x+h)/(b-(-4*
a*c+b^2)^(1/2)))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/i/(b-(-4*a*c+b^2)^(1/2))-g*polylog(2,-2*c*exp(i*x+h)/(b-(
-4*a*c+b^2)^(1/2)))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/i^2/(b-(-4*a*c+b^2)^(1/2))+1/2*(g*x+f)^2*(e+(b*e-2*c*d
)/(-4*a*c+b^2)^(1/2))/g/(b+(-4*a*c+b^2)^(1/2))-(g*x+f)*ln(1+2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))*(e+(b*e-2*c
*d)/(-4*a*c+b^2)^(1/2))/i/(b+(-4*a*c+b^2)^(1/2))-g*polylog(2,-2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))*(e+(b*e-2
*c*d)/(-4*a*c+b^2)^(1/2))/i^2/(b+(-4*a*c+b^2)^(1/2))

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2297, 2215, 2221, 2317, 2438} \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=-\frac {(f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{2 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{2 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {g \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {g \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )} \]

[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 2215

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2297

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 2317

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 2438

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*} \text {integral}& = -\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+i 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+i 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+i x} (f+g x)}{b+\sqrt {b^2-4 a c}+2 c e^{h+i 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+i x} (f+g x)}{b-\sqrt {b^2-4 a c}+2 c e^{h+i 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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}+\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+i x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b+\sqrt {b^2-4 a c}\right ) i}+\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+i x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b-\sqrt {b^2-4 a c}\right ) i} \\ & = \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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}+\frac {\left (\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+i x}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {\left (\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+i x}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2} \\ & = \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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=-\frac {i \left (-\sqrt {-\left (b^2-4 a c\right )^2} d g i x^2+2 \sqrt {b^2-4 a c} (b d-2 a e) f \arctan \left (\frac {b+2 c e^{h+i x}}{\sqrt {-b^2+4 a c}}\right )-2 \sqrt {-\left (b^2-4 a c\right )^2} d f \log \left (e^{h+i x}\right )+\sqrt {-\left (b^2-4 a c\right )^2} d g x \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )+b \sqrt {-b^2+4 a c} d g x \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )-2 a \sqrt {-b^2+4 a c} e g x \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )+\sqrt {-\left (b^2-4 a c\right )^2} d g x \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )-b \sqrt {-b^2+4 a c} d g x \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+2 a \sqrt {-b^2+4 a c} e g x \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+\sqrt {-\left (b^2-4 a c\right )^2} d f \log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right )\right )+\left (\sqrt {-\left (b^2-4 a c\right )^2} d+b \sqrt {-b^2+4 a c} d-2 a \sqrt {-b^2+4 a c} e\right ) g \operatorname {PolyLog}\left (2,\frac {2 c e^{h+i x}}{-b+\sqrt {b^2-4 a c}}\right )+\left (\sqrt {-\left (b^2-4 a c\right )^2} d-b \sqrt {-b^2+4 a c} d+2 a \sqrt {-b^2+4 a c} e\right ) g \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {-\left (b^2-4 a c\right )^2} i^2} \]

[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]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1181\) vs. \(2(382)=764\).

Time = 0.12 (sec) , antiderivative size = 1182, normalized size of antiderivative = 2.76

method	result	size
default	\(\text {Expression too large to display}\)	\(1182\)
risch	\(\text {Expression too large to display}\)	\(1319\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.52 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {{\left (b^{2} - 4 \, a c\right )} d g i^{2} x^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} d f i^{2} x - {\left ({\left (b^{2} - 4 \, a c\right )} d g + {\left (a b d - 2 \, a^{2} e\right )} g \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} {\rm Li}_2\left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} + b e^{\left (i x + h\right )} + 2 \, a}{2 \, a} + 1\right ) - {\left ({\left (b^{2} - 4 \, a c\right )} d g - {\left (a b d - 2 \, a^{2} e\right )} g \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} {\rm Li}_2\left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} - b e^{\left (i x + h\right )} - 2 \, a}{2 \, a} + 1\right ) + {\left ({\left (b^{2} - 4 \, a c\right )} d g h - {\left (b^{2} - 4 \, a c\right )} d f i - {\left ({\left (a b d - 2 \, a^{2} e\right )} g h - {\left (a b d - 2 \, a^{2} e\right )} f i\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (2 \, c e^{\left (i x + h\right )} + a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) + {\left ({\left (b^{2} - 4 \, a c\right )} d g h - {\left (b^{2} - 4 \, a c\right )} d f i + {\left ({\left (a b d - 2 \, a^{2} e\right )} g h - {\left (a b d - 2 \, a^{2} e\right )} f i\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (2 \, c e^{\left (i x + h\right )} - a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - {\left ({\left (b^{2} - 4 \, a c\right )} d g i x + {\left (b^{2} - 4 \, a c\right )} d g h + {\left ({\left (a b d - 2 \, a^{2} e\right )} g i x + {\left (a b d - 2 \, a^{2} e\right )} g h\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} + b e^{\left (i x + h\right )} + 2 \, a}{2 \, a}\right ) - {\left ({\left (b^{2} - 4 \, a c\right )} d g i x + {\left (b^{2} - 4 \, a c\right )} d g h - {\left ({\left (a b d - 2 \, a^{2} e\right )} g i x + {\left (a b d - 2 \, a^{2} e\right )} g h\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} - b e^{\left (i x + h\right )} - 2 \, a}{2 \, a}\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} i^{2}} \]

[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)

Sympy [F]

\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int \frac {\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Exception raised: ValueError} \]

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [F]

\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (d+e\,{\mathrm {e}}^{h+i\,x}\right )}{a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}} \,d x \]

[In]

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

[Out]

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