3.85.31 \(\int \frac {e^{2 x} (-2+2 e^5)+e^{2 e^5 x} (8 e^{8 x}+2 x+e^{4 x} (2+8 x))}{-e^{2 x}+e^{2 e^5 x} (-4+e^{8 x}+2 e^{4 x} x+x^2)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (-4-e^{2 x-2 e^5 x}+\left (e^{4 x}+x\right )^2\right ) \]

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Rubi [F]  time = 13.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(-2 + 2*E^5) + E^(2*E^5*x)*(8*E^(8*x) + 2*x + E^(4*x)*(2 + 8*x)))/(-E^(2*x) + E^(2*E^5*x)*(-4 + E
^(8*x) + 2*E^(4*x)*x + x^2)),x]

[Out]

Log[4 - E^(8*x) - 2*E^(4*x)*x - x^2] - 8*(1 - E^5)*Defer[Int][E^(2*x)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(
2*x) + 4*E^(2*E^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E^(2*E^5*x)*x^2)), x] - 2*Defer[Int][E^(6*x
)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(2*x) + 4*E^(2*E^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E
^(2*E^5*x)*x^2)), x] - 2*(3 + E^5)*Defer[Int][E^(10*x)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(2*x) + 4*E^(2*E
^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E^(2*E^5*x)*x^2)), x] - 4*(1 + E^5)*Defer[Int][(E^(6*x)*x)
/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(2*x) + 4*E^(2*E^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E^
(2*E^5*x)*x^2)), x] + 2*Defer[Int][(E^(2*x)*x)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(-E^(2*x) - 4*E^(2*E^5*x) +
E^(2*(4 + E^5)*x) + 2*E^(2*(2 + E^5)*x)*x + E^(2*E^5*x)*x^2)), x] - 2*(1 - E^5)*Defer[Int][(E^(2*x)*x^2)/((-4
+ E^(8*x) + 2*E^(4*x)*x + x^2)*(-E^(2*x) - 4*E^(2*E^5*x) + E^(2*(4 + E^5)*x) + 2*E^(2*(2 + E^5)*x)*x + E^(2*E
^5*x)*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (1+4 e^{4 x}\right ) \left (e^{4 x}+x\right )}{-4+e^{8 x}+2 e^{4 x} x+x^2}+\frac {2 e^{2 x} \left (e^{4 x}+4 \left (1-e^5\right )+3 e^{8 x} \left (1+\frac {e^5}{3}\right )+x+2 e^{4 x} \left (1+e^5\right ) x-\left (1-e^5\right ) x^2\right )}{\left (4-e^{8 x}-2 e^{4 x} x-x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )}\right ) \, dx\\ &=2 \int \frac {\left (1+4 e^{4 x}\right ) \left (e^{4 x}+x\right )}{-4+e^{8 x}+2 e^{4 x} x+x^2} \, dx+2 \int \frac {e^{2 x} \left (e^{4 x}+4 \left (1-e^5\right )+3 e^{8 x} \left (1+\frac {e^5}{3}\right )+x+2 e^{4 x} \left (1+e^5\right ) x-\left (1-e^5\right ) x^2\right )}{\left (4-e^{8 x}-2 e^{4 x} x-x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )} \, dx\\ &=\log \left (4-e^{8 x}-2 e^{4 x} x-x^2\right )+2 \int \left (-\frac {e^{6 x}}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )}+\frac {4 e^{2 x} \left (-1+e^5\right )}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )}-\frac {e^{10 x} \left (3+e^5\right )}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )}-\frac {2 e^{6 x} \left (1+e^5\right ) x}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )}+\frac {e^{2 x} x}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}+2 e^{2 \left (2+e^5\right ) x} x+e^{2 e^5 x} x^2\right )}+\frac {e^{2 x} \left (-1+e^5\right ) x^2}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}+2 e^{2 \left (2+e^5\right ) x} x+e^{2 e^5 x} x^2\right )}\right ) \, dx\\ &=\log \left (4-e^{8 x}-2 e^{4 x} x-x^2\right )-2 \int \frac {e^{6 x}}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )} \, dx+2 \int \frac {e^{2 x} x}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}+2 e^{2 \left (2+e^5\right ) x} x+e^{2 e^5 x} x^2\right )} \, dx-\left (2 \left (1-e^5\right )\right ) \int \frac {e^{2 x} x^2}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}+2 e^{2 \left (2+e^5\right ) x} x+e^{2 e^5 x} x^2\right )} \, dx-\left (8 \left (1-e^5\right )\right ) \int \frac {e^{2 x}}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )} \, dx-\left (4 \left (1+e^5\right )\right ) \int \frac {e^{6 x} x}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )} \, dx-\left (2 \left (3+e^5\right )\right ) \int \frac {e^{10 x}}{\left (-4+e^{8 x}+2 e^{4 x} x+x^2\right ) \left (e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}-2 e^{2 \left (2+e^5\right ) x} x-e^{2 e^5 x} x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 5.13, size = 65, normalized size = 2.50 \begin {gather*} -2 e^5 x+\log \left (-e^{2 x}-4 e^{2 e^5 x}+e^{8 x+2 e^5 x}+2 e^{4 x+2 e^5 x} x+e^{2 e^5 x} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-2 + 2*E^5) + E^(2*E^5*x)*(8*E^(8*x) + 2*x + E^(4*x)*(2 + 8*x)))/(-E^(2*x) + E^(2*E^5*x)*(
-4 + E^(8*x) + 2*E^(4*x)*x + x^2)),x]

[Out]

-2*E^5*x + Log[-E^(2*x) - 4*E^(2*E^5*x) + E^(8*x + 2*E^5*x) + 2*E^(4*x + 2*E^5*x)*x + E^(2*E^5*x)*x^2]

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fricas [B]  time = 1.91, size = 73, normalized size = 2.81 \begin {gather*} -2 \, x e^{5} + \log \left (x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4\right ) + \log \left (\frac {{\left (x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4\right )} e^{\left (2 \, x e^{5}\right )} - e^{\left (2 \, x\right )}}{x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x
)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="fricas")

[Out]

-2*x*e^5 + log(x^2 + 2*x*e^(4*x) + e^(8*x) - 4) + log(((x^2 + 2*x*e^(4*x) + e^(8*x) - 4)*e^(2*x*e^5) - e^(2*x)
)/(x^2 + 2*x*e^(4*x) + e^(8*x) - 4))

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giac [B]  time = 0.42, size = 55, normalized size = 2.12 \begin {gather*} -2 \, x e^{5} + \log \left (x^{2} e^{\left (2 \, x e^{5}\right )} + 2 \, x e^{\left (2 \, x e^{5} + 4 \, x\right )} - 4 \, e^{\left (2 \, x e^{5}\right )} + e^{\left (2 \, x e^{5} + 8 \, x\right )} - e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x
)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="giac")

[Out]

-2*x*e^5 + log(x^2*e^(2*x*e^5) + 2*x*e^(2*x*e^5 + 4*x) - 4*e^(2*x*e^5) + e^(2*x*e^5 + 8*x) - e^(2*x))

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maple [B]  time = 0.07, size = 56, normalized size = 2.15

method	result	size
risch	\(\ln \left ({\mathrm e}^{8 x}+2 x \,{\mathrm e}^{4 x}+x^{2}-4\right )-2 x \,{\mathrm e}^{5}+\ln \left ({\mathrm e}^{2 x \,{\mathrm e}^{5}}-\frac {{\mathrm e}^{2 x}}{{\mathrm e}^{8 x}+2 x \,{\mathrm e}^{4 x}+x^{2}-4}\right )\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-
4)*exp(x*exp(5))^2-exp(x)^2),x,method=_RETURNVERBOSE)

[Out]

ln(exp(8*x)+2*x*exp(4*x)+x^2-4)-2*x*exp(5)+ln(exp(2*x*exp(5))-exp(2*x)/(exp(8*x)+2*x*exp(4*x)+x^2-4))

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maxima [B]  time = 0.41, size = 72, normalized size = 2.77 \begin {gather*} -2 \, x e^{5} + \log \left (x + e^{\left (4 \, x\right )} + 2\right ) + \log \left (x + e^{\left (4 \, x\right )} - 2\right ) + \log \left (\frac {{\left (x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4\right )} e^{\left (2 \, x e^{5}\right )} - e^{\left (2 \, x\right )}}{x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x
)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="maxima")

[Out]

-2*x*e^5 + log(x + e^(4*x) + 2) + log(x + e^(4*x) - 2) + log(((x^2 + 2*x*e^(4*x) + e^(8*x) - 4)*e^(2*x*e^5) -
e^(2*x))/(x^2 + 2*x*e^(4*x) + e^(8*x) - 4))

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mupad [B]  time = 0.58, size = 92, normalized size = 3.54 \begin {gather*} \ln \left (\frac {{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}-4\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}-{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}+2\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}}{{\mathrm {e}}^{8\,x}+2\,x\,{\mathrm {e}}^{4\,x}+x^2-4}\right )+\ln \left ({\mathrm {e}}^{8\,x}+2\,x\,{\mathrm {e}}^{4\,x}+x^2-4\right )-2\,x\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x*exp(5))*(2*x + 8*exp(8*x) + exp(4*x)*(8*x + 2)) + exp(2*x)*(2*exp(5) - 2))/(exp(2*x) - exp(2*x*e
xp(5))*(exp(8*x) + 2*x*exp(4*x) + x^2 - 4)),x)

[Out]

log((exp(8*x)*exp(2*x*exp(5)) - 4*exp(2*x*exp(5)) - exp(2*x) + x^2*exp(2*x*exp(5)) + 2*x*exp(4*x)*exp(2*x*exp(
5)))/(exp(8*x) + 2*x*exp(4*x) + x^2 - 4)) + log(exp(8*x) + 2*x*exp(4*x) + x^2 - 4) - 2*x*exp(5)

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sympy [B]  time = 2.89, size = 65, normalized size = 2.50 \begin {gather*} - 2 x e^{5} + \log {\left (x^{2} e^{2 x e^{5}} + 2 x e^{4 x} e^{2 x e^{5}} + e^{8 x} e^{2 x e^{5}} - e^{2 x} - 4 e^{2 x e^{5}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)**2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))**2+(2*exp(5)-2)*exp(x)**2)/((exp(4*x)**2+2*x*exp
(4*x)+x**2-4)*exp(x*exp(5))**2-exp(x)**2),x)

[Out]

-2*x*exp(5) + log(x**2*exp(2*x*exp(5)) + 2*x*exp(4*x)*exp(2*x*exp(5)) + exp(8*x)*exp(2*x*exp(5)) - exp(2*x) -
4*exp(2*x*exp(5)))

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