3.6.76 \(\int \frac {d+e e^{h+i x}}{(a+b e^{h+i x}+c e^{2 h+2 i x}) (f+g x)} \, dx\) [576]

Optimal. Leaf size=84 \[ d \text {Int}\left (\frac {1}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)},x\right )+e \text {Int}\left (\frac {e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)},x\right ) \]

[Out]

d*CannotIntegrate(1/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h))/(g*x+f),x)+e*CannotIntegrate(exp(i*x+h)/(a+b*exp(i*x+h)+
c*exp(2*i*x+2*h))/(g*x+f),x)

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Rubi [A]
time = 0.69, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

d*Defer[Int][1/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x] + e*Defer[Int][E^(h + i*x)/((a + b*E^(h
+ i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x]

Rubi steps

\begin {align*} \int \frac {d+e e^{h+576 x}}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)} \, dx &=\int \left (\frac {d}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)}+\frac {e e^{h+576 x}}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)}\right ) \, dx\\ &=d \int \frac {1}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)} \, dx+e \int \frac {e^{h+576 x}}{\left (a+b e^{h+576 x}+c e^{2 h+1152 x}\right ) (f+g x)} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e e^{h+i x}}{\left (a+b e^{h+i x}+c e^{2 h+2 i x}\right ) (f+g x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {d +e \,{\mathrm e}^{i x +h}}{\left (a +b \,{\mathrm e}^{i x +h}+c \,{\mathrm e}^{2 i x +2 h}\right ) \left (g x +f \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

integral((d*e^2 + e^(h + I*x + 3))/((a*g*x + a*f)*e^2 + (c*g*x + c*f)*e^(2*h + 2*I*x + 2) + (b*g*x + b*f)*e^(h
+ I*x + 2)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e e^{h} e^{i x}}{\left (f + g x\right ) \left (a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d + e^(h + I*x + 1))/((g*x + f)*(c*e^(2*h + 2*I*x) + b*e^(h + I*x) + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {d+e\,{\mathrm {e}}^{h+i\,x}}{\left (f+g\,x\right )\,\left (a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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