3.5 \(\int \frac{1}{\left (a+b e^{c+d x}\right ) x} \, dx\)

Optimal. Leaf size=20 \[ \text{Int}\left (\frac{1}{x \left (a+b e^{c+d x}\right )},x\right ) \]

[Out]

Unintegrable[1/((a + b*E^(c + d*x))*x), x]

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Rubi [A]  time = 0.0741346, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{1}{\left (a+b e^{c+d x}\right ) x},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[1/((a + b*E^(c + d*x))*x),x]

[Out]

Defer[Int][1/((a + b*E^(c + d*x))*x), x]

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b e^{c + d x}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*exp(d*x+c))/x,x)

[Out]

Integral(1/(x*(a + b*exp(c + d*x))), x)

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Mathematica [A]  time = 0.104162, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b e^{c+d x}\right ) x} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/((a + b*E^(c + d*x))*x),x]

[Out]

Integrate[1/((a + b*E^(c + d*x))*x), x]

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Maple [A]  time = 0.017, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{{\rm e}^{dx+c}} \right ) x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(d*x+c))/x,x)

[Out]

int(1/(a+b*exp(d*x+c))/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*e^(d*x + c) + a)*x),x, algorithm="maxima")

[Out]

integrate(1/((b*e^(d*x + c) + a)*x), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b x e^{\left (d x + c\right )} + a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*e^(d*x + c) + a)*x),x, algorithm="fricas")

[Out]

integral(1/(b*x*e^(d*x + c) + a*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b e^{c} e^{d x}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*exp(d*x+c))/x,x)

[Out]

Integral(1/(x*(a + b*exp(c)*exp(d*x))), x)

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*e^(d*x + c) + a)*x),x, algorithm="giac")

[Out]

integrate(1/((b*e^(d*x + c) + a)*x), x)