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

Optimal. Leaf size=22 \[ -\frac{1}{2 a p \left (a e^{2 p x}+b\right )} \]

[Out]

-1/(2*a*(b + a*E^(2*p*x))*p)

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Rubi [A]  time = 0.0413316, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{1}{2 a p \left (a e^{2 p x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a*(b + a*E^(2*p*x))*p)

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Rubi in Sympy [A]  time = 4.67805, size = 15, normalized size = 0.68 \[ \frac{1}{2 b p \left (a + b e^{- 2 p x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*b*p*(a + b*exp(-2*p*x)))

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Mathematica [A]  time = 0.0211906, size = 22, normalized size = 1. \[ -\frac{1}{2 a p \left (a e^{2 p x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a*(b + a*E^(2*p*x))*p)

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Maple [A]  time = 0.004, size = 21, normalized size = 1. \[ -{\frac{1}{2\,pa \left ( a \left ({{\rm e}^{px}} \right ) ^{2}+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34244, size = 27, normalized size = 1.23 \[ \frac{1}{2 \,{\left (b^{2} e^{\left (-2 \, p x\right )} + a b\right )} p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/((b^2*e^(-2*p*x) + a*b)*p)

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Fricas [A]  time = 0.215691, size = 26, normalized size = 1.18 \[ -\frac{1}{2 \,{\left (a^{2} p e^{\left (2 \, p x\right )} + a b p\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a^2*p*e^(2*p*x) + a*b*p)

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Sympy [A]  time = 0.116086, size = 22, normalized size = 1. \[ - \frac{1}{2 a^{2} p e^{2 p x} + 2 a b p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a**2*p*exp(2*p*x) + 2*a*b*p)

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GIAC/XCAS [A]  time = 0.200777, size = 26, normalized size = 1.18 \[ -\frac{1}{2 \,{\left (a e^{\left (2 \, p x\right )} + b\right )} a p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((a*e^(2*p*x) + b)*a*p)