3.1.13 \(\int \frac {1}{(a+b e^{c+d x})^2 x} \, dx\) [13]

Optimal. Leaf size=20 \[ \text {Int}\left (\frac {1}{\left (a+b e^{c+d x}\right )^2 x},x\right ) \]

[Out]

Unintegrable(1/(a+b*exp(d*x+c))^2/x,x)

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Rubi [A]
time = 0.03, 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 {1}{\left (a+b e^{c+d x}\right )^2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((a + b*E^(c + d*x))^2*x),x]

[Out]

Defer[Int][1/((a + b*E^(c + d*x))^2*x), x]

Rubi steps

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

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Mathematica [A]
time = 1.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b e^{c+d x}\right )^2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((a + b*E^(c + d*x))^2*x),x]

[Out]

Integrate[1/((a + b*E^(c + d*x))^2*x), x]

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Maple [A]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,{\mathrm e}^{d x +c}\right )^{2} x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(d*x+c))^2/x,x)

[Out]

int(1/(a+b*exp(d*x+c))^2/x,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(1/(a+b*exp(d*x+c))^2/x,x, algorithm="maxima")

[Out]

1/(a*b*d*x*e^(d*x + c) + a^2*d*x) + integrate((d*x + 1)/(a*b*d*x^2*e^(d*x + c) + a^2*d*x^2), 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(1/(a+b*exp(d*x+c))^2/x,x, algorithm="fricas")

[Out]

integral(1/(b^2*x*e^(2*d*x + 2*c) + 2*a*b*x*e^(d*x + c) + a^2*x), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{a^{2} d x + a b d x e^{c + d x}} + \frac {\int \frac {d x}{a x^{2} + b x^{2} e^{c} e^{d x}}\, dx + \int \frac {1}{a x^{2} + b x^{2} e^{c} e^{d x}}\, dx}{a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))**2/x,x)

[Out]

1/(a**2*d*x + a*b*d*x*exp(c + d*x)) + (Integral(d*x/(a*x**2 + b*x**2*exp(c)*exp(d*x)), x) + Integral(1/(a*x**2
+ b*x**2*exp(c)*exp(d*x)), x))/(a*d)

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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(1/(a+b*exp(d*x+c))^2/x,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{x\,{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*exp(c + d*x))^2),x)

[Out]

int(1/(x*(a + b*exp(c + d*x))^2), x)

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