3.649 \(\int \frac{e^{5 x}}{1+e^{10 x}} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{5} \tan ^{-1}\left (e^{5 x}\right ) \]

[Out]

ArcTan[E^(5*x)]/5

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Rubi [A]  time = 0.0328427, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{5} \tan ^{-1}\left (e^{5 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(5*x)/(1 + E^(10*x)),x]

[Out]

ArcTan[E^(5*x)]/5

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Rubi in Sympy [A]  time = 5.89509, size = 7, normalized size = 0.7 \[ \frac{\operatorname{atan}{\left (e^{5 x} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(5*x)/(1+exp(10*x)),x)

[Out]

atan(exp(5*x))/5

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Mathematica [A]  time = 0.00634014, size = 10, normalized size = 1. \[ \frac{1}{5} \tan ^{-1}\left (e^{5 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(5*x)/(1 + E^(10*x)),x]

[Out]

ArcTan[E^(5*x)]/5

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Maple [A]  time = 0.006, size = 8, normalized size = 0.8 \[{\frac{\arctan \left ( \left ({{\rm e}^{x}} \right ) ^{5} \right ) }{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(5*x)/(1+exp(10*x)),x)

[Out]

1/5*arctan(exp(x)^5)

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Maxima [A]  time = 0.868714, size = 9, normalized size = 0.9 \[ \frac{1}{5} \, \arctan \left (e^{\left (5 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(5*x)/(e^(10*x) + 1),x, algorithm="maxima")

[Out]

1/5*arctan(e^(5*x))

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Fricas [A]  time = 0.246733, size = 9, normalized size = 0.9 \[ \frac{1}{5} \, \arctan \left (e^{\left (5 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(5*x)/(e^(10*x) + 1),x, algorithm="fricas")

[Out]

1/5*arctan(e^(5*x))

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Sympy [A]  time = 0.102071, size = 17, normalized size = 1.7 \[ \operatorname{RootSum}{\left (100 z^{2} + 1, \left ( i \mapsto i \log{\left (10 i + e^{5 x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(5*x)/(1+exp(10*x)),x)

[Out]

RootSum(100*_z**2 + 1, Lambda(_i, _i*log(10*_i + exp(5*x))))

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GIAC/XCAS [A]  time = 0.235747, size = 9, normalized size = 0.9 \[ \frac{1}{5} \, \arctan \left (e^{\left (5 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(5*x)/(e^(10*x) + 1),x, algorithm="giac")

[Out]

1/5*arctan(e^(5*x))