3.11 \(\int \frac{x}{\left (a+b e^{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^2}+\frac{\log \left (a+b e^{c+d x}\right )}{a^2 d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^2 d}-\frac{x}{a^2 d}+\frac{x^2}{2 a^2}+\frac{x}{a d \left (a+b e^{c+d x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.324003, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^2}+\frac{\log \left (a+b e^{c+d x}\right )}{a^2 d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^2 d}-\frac{x}{a^2 d}+\frac{x^2}{2 a^2}+\frac{x}{a d \left (a+b e^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*E^(c + d*x))^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x e^{- c - d x} e^{c + d x}}{a d \left (a + b e^{c + d x}\right )} + \frac{\int x\, dx}{a^{2}} - \frac{x e^{- c - d x} e^{c + d x} \log{\left (a + b e^{c + d x} \right )}}{a^{2} d} + \frac{x e^{- c - d x} e^{c + d x} \log{\left (e^{c + d x} \right )}}{a^{2} d} - \frac{x \log{\left (1 + \frac{b e^{c + d x}}{a} \right )}}{a^{2} d} + \frac{x \log{\left (a + b e^{c + d x} \right )}}{a^{2} d} - \frac{x \log{\left (e^{c + d x} \right )}}{a^{2} d} + \frac{\log{\left (a + b e^{c + d x} \right )}}{a^{2} d^{2}} - \frac{\log{\left (e^{c + d x} \right )}}{a^{2} d^{2}} - \frac{\operatorname{Li}_{2}\left (- \frac{b e^{c + d x}}{a}\right )}{a^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*exp(d*x+c))**2,x)

[Out]

x*exp(-c - d*x)*exp(c + d*x)/(a*d*(a + b*exp(c + d*x))) + Integral(x, x)/a**2 -
x*exp(-c - d*x)*exp(c + d*x)*log(a + b*exp(c + d*x))/(a**2*d) + x*exp(-c - d*x)*
exp(c + d*x)*log(exp(c + d*x))/(a**2*d) - x*log(1 + b*exp(c + d*x)/a)/(a**2*d) +
x*log(a + b*exp(c + d*x))/(a**2*d) - x*log(exp(c + d*x))/(a**2*d) + log(a + b*e
xp(c + d*x))/(a**2*d**2) - log(exp(c + d*x))/(a**2*d**2) - polylog(2, -b*exp(c +
d*x)/a)/(a**2*d**2)

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Mathematica [A]  time = 0.153561, size = 85, normalized size = 0.79 \[ \frac{-2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )+\frac{d x \left (a d x+b (d x-2) e^{c+d x}\right )}{a+b e^{c+d x}}-2 (d x-1) \log \left (\frac{b e^{c+d x}}{a}+1\right )}{2 a^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.041, size = 231, normalized size = 2.2 \[{\frac{{x}^{2}}{2\,{a}^{2}}}+{\frac{xc}{d{a}^{2}}}+{\frac{{c}^{2}}{2\,{d}^{2}{a}^{2}}}+{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{d}^{2}{a}^{2}}}-{\frac{b{{\rm e}^{dx+c}}x}{d{a}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{b{{\rm e}^{dx+c}}c}{{d}^{2}{a}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{1}{{d}^{2}{a}^{2}}{\it dilog} \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{x}{d{a}^{2}}\ln \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c}{{d}^{2}{a}^{2}}\ln \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c\ln \left ({{\rm e}^{dx+c}} \right ) }{{d}^{2}{a}^{2}}}+{\frac{c\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{d}^{2}{a}^{2}}}-{\frac{c}{{d}^{2}a \left ( a+b{{\rm e}^{dx+c}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*exp(d*x+c))^2,x)

[Out]

1/2*x^2/a^2+1/d/a^2*x*c+1/2/d^2/a^2*c^2+ln(a+b*exp(d*x+c))/a^2/d^2-1/d/a^2*b*exp
(d*x+c)/(a+b*exp(d*x+c))*x-1/d^2/a^2*b*exp(d*x+c)/(a+b*exp(d*x+c))*c-1/d^2/a^2*d
ilog((a+b*exp(d*x+c))/a)-1/d/a^2*ln((a+b*exp(d*x+c))/a)*x-1/d^2/a^2*ln((a+b*exp(
d*x+c))/a)*c-1/d^2*c/a^2*ln(exp(d*x+c))+1/d^2*c/a^2*ln(a+b*exp(d*x+c))-1/d^2*c/a
/(a+b*exp(d*x+c))

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Maxima [A]  time = 0.791863, size = 128, normalized size = 1.2 \[ \frac{x}{a b d e^{\left (d x + c\right )} + a^{2} d} + \frac{x^{2}}{2 \, a^{2}} - \frac{x}{a^{2} d} - \frac{d x \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right )}{a^{2} d^{2}} + \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*e^(d*x + c) + a)^2,x, algorithm="maxima")

[Out]

x/(a*b*d*e^(d*x + c) + a^2*d) + 1/2*x^2/a^2 - x/(a^2*d) - (d*x*log(b*e^(d*x + c)
/a + 1) + dilog(-b*e^(d*x + c)/a))/(a^2*d^2) + log(b*e^(d*x + c) + a)/(a^2*d^2)

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Fricas [A]  time = 0.255239, size = 238, normalized size = 2.22 \[ \frac{a d^{2} x^{2} - a c^{2} - 2 \, a c - 2 \,{\left (b e^{\left (d x + c\right )} + a\right )}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) +{\left (b d^{2} x^{2} - b c^{2} - 2 \, b d x - 2 \, b c\right )} e^{\left (d x + c\right )} + 2 \,{\left (a c +{\left (b c + b\right )} e^{\left (d x + c\right )} + a\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \,{\left (a d x + a c +{\left (b d x + b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right )}{2 \,{\left (a^{2} b d^{2} e^{\left (d x + c\right )} + a^{3} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*e^(d*x + c) + a)^2,x, algorithm="fricas")

[Out]

1/2*(a*d^2*x^2 - a*c^2 - 2*a*c - 2*(b*e^(d*x + c) + a)*dilog(-(b*e^(d*x + c) + a
)/a + 1) + (b*d^2*x^2 - b*c^2 - 2*b*d*x - 2*b*c)*e^(d*x + c) + 2*(a*c + (b*c + b
)*e^(d*x + c) + a)*log(b*e^(d*x + c) + a) - 2*(a*d*x + a*c + (b*d*x + b*c)*e^(d*
x + c))*log((b*e^(d*x + c) + a)/a))/(a^2*b*d^2*e^(d*x + c) + a^3*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x}{a^{2} d + a b d e^{c + d x}} + \frac{\int \frac{d x}{a + b e^{c} e^{d x}}\, dx + \int \left (- \frac{1}{a + b e^{c} e^{d x}}\right )\, dx}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*exp(d*x+c))**2,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b e^{\left (d x + c\right )} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*e^(d*x + c) + a)^2,x, algorithm="giac")

[Out]

integrate(x/(b*e^(d*x + c) + a)^2, x)