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