3.21 \(\int \frac{1}{\left (a+b e^{p x}\right )^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{\log \left (a+b e^{p x}\right )}{a^2 p}+\frac{x}{a^2}+\frac{1}{a p \left (a+b e^{p x}\right )} \]

[Out]

1/(a*(a + b*E^(p*x))*p) + x/a^2 - Log[a + b*E^(p*x)]/(a^2*p)

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Rubi [A]  time = 0.0631128, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\log \left (a+b e^{p x}\right )}{a^2 p}+\frac{x}{a^2}+\frac{1}{a p \left (a+b e^{p x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^(p*x))^(-2),x]

[Out]

1/(a*(a + b*E^(p*x))*p) + x/a^2 - Log[a + b*E^(p*x)]/(a^2*p)

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Rubi in Sympy [A]  time = 5.39135, size = 39, normalized size = 0.93 \[ \frac{1}{a p \left (a + b e^{p x}\right )} - \frac{\log{\left (a + b e^{p x} \right )}}{a^{2} p} + \frac{\log{\left (e^{p x} \right )}}{a^{2} p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*exp(p*x))**2,x)

[Out]

1/(a*p*(a + b*exp(p*x))) - log(a + b*exp(p*x))/(a**2*p) + log(exp(p*x))/(a**2*p)

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Mathematica [A]  time = 0.0630322, size = 37, normalized size = 0.88 \[ \frac{\frac{a}{a p+b p e^{p x}}-\frac{\log \left (a+b e^{p x}\right )}{p}+x}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^(p*x))^(-2),x]

[Out]

(a/(a*p + b*E^(p*x)*p) + x - Log[a + b*E^(p*x)]/p)/a^2

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Maple [A]  time = 0.017, size = 48, normalized size = 1.1 \[ -{\frac{\ln \left ( a+b{{\rm e}^{px}} \right ) }{{a}^{2}p}}+{\frac{1}{a \left ( a+b{{\rm e}^{px}} \right ) p}}+{\frac{\ln \left ({{\rm e}^{px}} \right ) }{{a}^{2}p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(p*x))^2,x)

[Out]

-ln(a+b*exp(p*x))/a^2/p+1/a/(a+b*exp(p*x))/p+1/p/a^2*ln(exp(p*x))

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Maxima [A]  time = 1.35229, size = 54, normalized size = 1.29 \[ \frac{x}{a^{2}} + \frac{1}{{\left (a b e^{\left (p x\right )} + a^{2}\right )} p} - \frac{\log \left (b e^{\left (p x\right )} + a\right )}{a^{2} p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(p*x) + a)^(-2),x, algorithm="maxima")

[Out]

x/a^2 + 1/((a*b*e^(p*x) + a^2)*p) - log(b*e^(p*x) + a)/(a^2*p)

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Fricas [A]  time = 0.224159, size = 70, normalized size = 1.67 \[ \frac{b p x e^{\left (p x\right )} + a p x -{\left (b e^{\left (p x\right )} + a\right )} \log \left (b e^{\left (p x\right )} + a\right ) + a}{a^{2} b p e^{\left (p x\right )} + a^{3} p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(p*x) + a)^(-2),x, algorithm="fricas")

[Out]

(b*p*x*e^(p*x) + a*p*x - (b*e^(p*x) + a)*log(b*e^(p*x) + a) + a)/(a^2*b*p*e^(p*x
) + a^3*p)

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Sympy [A]  time = 0.17583, size = 36, normalized size = 0.86 \[ \frac{1}{a^{2} p + a b p e^{p x}} + \frac{x}{a^{2}} - \frac{\log{\left (\frac{a}{b} + e^{p x} \right )}}{a^{2} p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*exp(p*x))**2,x)

[Out]

1/(a**2*p + a*b*p*exp(p*x)) + x/a**2 - log(a/b + exp(p*x))/(a**2*p)

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GIAC/XCAS [A]  time = 0.197701, size = 55, normalized size = 1.31 \[ \frac{x}{a^{2}} - \frac{{\rm ln}\left ({\left | b e^{\left (p x\right )} + a \right |}\right )}{a^{2} p} + \frac{1}{{\left (b e^{\left (p x\right )} + a\right )} a p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(p*x) + a)^(-2),x, algorithm="giac")

[Out]

x/a^2 - ln(abs(b*e^(p*x) + a))/(a^2*p) + 1/((b*e^(p*x) + a)*a*p)