3.22 \(\int \frac{e^{4 x}}{a+b e^{2 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0609833, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[E^(4*x)/(a + b*E^(2*x)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(4*x)/(a+b*exp(2*x)),x)

[Out]

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

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Mathematica [A]  time = 0.0154846, size = 28, normalized size = 0.9 \[ \frac{b e^{2 x}-a \log \left (a+b e^{2 x}\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(4*x)/(a + b*E^(2*x)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 26, normalized size = 0.8 \[{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2\,b}}-{\frac{a\ln \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(4*x)/(a+b*exp(2*x)),x)

[Out]

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

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Maxima [A]  time = 0.776565, size = 34, normalized size = 1.1 \[ \frac{e^{\left (2 \, x\right )}}{2 \, b} - \frac{a \log \left (b e^{\left (2 \, x\right )} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(4*x)/(b*e^(2*x) + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.249887, size = 32, normalized size = 1.03 \[ \frac{b e^{\left (2 \, x\right )} - a \log \left (b e^{\left (2 \, x\right )} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(4*x)/(b*e^(2*x) + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.284495, size = 29, normalized size = 0.94 \[ - \frac{a \log{\left (\frac{a}{b} + e^{2 x} \right )}}{2 b^{2}} + \begin{cases} \frac{e^{2 x}}{2 b} & \text{for}\: 2 b \neq 0 \\\frac{x}{b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(4*x)/(a+b*exp(2*x)),x)

[Out]

-a*log(a/b + exp(2*x))/(2*b**2) + Piecewise((exp(2*x)/(2*b), Ne(2*b, 0)), (x/b,
True))

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GIAC/XCAS [A]  time = 0.248446, size = 35, normalized size = 1.13 \[ \frac{e^{\left (2 \, x\right )}}{2 \, b} - \frac{a{\rm ln}\left ({\left | b e^{\left (2 \, x\right )} + a \right |}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(4*x)/(b*e^(2*x) + a),x, algorithm="giac")

[Out]

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