3.103.12 \(\int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x (2 e^4-75 x-2 x^2)}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x (2 e^4 x-2 x^3)} \, dx\)

Optimal. Leaf size=28 \[ 4+\frac {75}{e^4+e^x-x^2}-\log (4)-\log \left (\frac {1}{x}\right ) \]

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Rubi [F]  time = 0.80, 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^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^8 + E^(2*x) + 150*x^2 - 2*E^4*x^2 + x^4 + E^x*(2*E^4 - 75*x - 2*x^2))/(E^8*x + E^(2*x)*x - 2*E^4*x^3 +
x^5 + E^x*(2*E^4*x - 2*x^3)),x]

[Out]

Log[x] + 75*E^4*Defer[Int][(E^4 + E^x - x^2)^(-2), x] - 75*Defer[Int][(E^4 + E^x - x^2)^(-1), x] + 150*Defer[I
nt][x/(-E^4 - E^x + x^2)^2, x] - 75*Defer[Int][x^2/(-E^4 - E^x + x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^8+e^{2 x}+\left (150-2 e^4\right ) x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx\\ &=\int \frac {e^8+e^{2 x}+2 e^{4+x}-2 e^4 x^2-e^x x (75+2 x)+x^2 \left (150+x^2\right )}{x \left (e^4+e^x-x^2\right )^2} \, dx\\ &=\int \left (\frac {1}{x}-\frac {75}{e^4+e^x-x^2}+\frac {75 \left (e^4+2 x-x^2\right )}{\left (e^4+e^x-x^2\right )^2}\right ) \, dx\\ &=\log (x)-75 \int \frac {1}{e^4+e^x-x^2} \, dx+75 \int \frac {e^4+2 x-x^2}{\left (e^4+e^x-x^2\right )^2} \, dx\\ &=\log (x)-75 \int \frac {1}{e^4+e^x-x^2} \, dx+75 \int \left (\frac {e^4}{\left (e^4+e^x-x^2\right )^2}+\frac {2 x}{\left (-e^4-e^x+x^2\right )^2}-\frac {x^2}{\left (-e^4-e^x+x^2\right )^2}\right ) \, dx\\ &=\log (x)-75 \int \frac {1}{e^4+e^x-x^2} \, dx-75 \int \frac {x^2}{\left (-e^4-e^x+x^2\right )^2} \, dx+150 \int \frac {x}{\left (-e^4-e^x+x^2\right )^2} \, dx+\left (75 e^4\right ) \int \frac {1}{\left (e^4+e^x-x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 19, normalized size = 0.68 \begin {gather*} \frac {75}{e^4+e^x-x^2}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^8 + E^(2*x) + 150*x^2 - 2*E^4*x^2 + x^4 + E^x*(2*E^4 - 75*x - 2*x^2))/(E^8*x + E^(2*x)*x - 2*E^4*
x^3 + x^5 + E^x*(2*E^4*x - 2*x^3)),x]

[Out]

75/(E^4 + E^x - x^2) + Log[x]

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fricas [A]  time = 0.57, size = 32, normalized size = 1.14 \begin {gather*} \frac {{\left (x^{2} - e^{4} - e^{x}\right )} \log \relax (x) - 75}{x^{2} - e^{4} - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*
x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^5),x, algorithm="fricas")

[Out]

((x^2 - e^4 - e^x)*log(x) - 75)/(x^2 - e^4 - e^x)

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giac [A]  time = 0.25, size = 35, normalized size = 1.25 \begin {gather*} \frac {x^{2} \log \relax (x) - e^{4} \log \relax (x) - e^{x} \log \relax (x) - 75}{x^{2} - e^{4} - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*
x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^5),x, algorithm="giac")

[Out]

(x^2*log(x) - e^4*log(x) - e^x*log(x) - 75)/(x^2 - e^4 - e^x)

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maple [A]  time = 0.17, size = 18, normalized size = 0.64

method	result	size
norman	\(\frac {75}{-x^{2}+{\mathrm e}^{4}+{\mathrm e}^{x}}+\ln \relax (x )\)	\(18\)
risch	\(\frac {75}{-x^{2}+{\mathrm e}^{4}+{\mathrm e}^{x}}+\ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*x^3)*e
xp(x)+x*exp(4)^2-2*x^3*exp(4)+x^5),x,method=_RETURNVERBOSE)

[Out]

75/(-x^2+exp(4)+exp(x))+ln(x)

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maxima [A]  time = 0.39, size = 19, normalized size = 0.68 \begin {gather*} -\frac {75}{x^{2} - e^{4} - e^{x}} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*
x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^5),x, algorithm="maxima")

[Out]

-75/(x^2 - e^4 - e^x) + log(x)

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mupad [B]  time = 9.55, size = 17, normalized size = 0.61 \begin {gather*} \ln \relax (x)+\frac {75}{{\mathrm {e}}^4+{\mathrm {e}}^x-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x) + exp(8) - exp(x)*(75*x - 2*exp(4) + 2*x^2) - 2*x^2*exp(4) + 150*x^2 + x^4)/(exp(x)*(2*x*exp(4)
- 2*x^3) + x*exp(2*x) + x*exp(8) - 2*x^3*exp(4) + x^5),x)

[Out]

log(x) + 75/(exp(4) + exp(x) - x^2)

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sympy [A]  time = 0.14, size = 14, normalized size = 0.50 \begin {gather*} \log {\relax (x )} + \frac {75}{- x^{2} + e^{x} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)**2+(2*exp(4)-2*x**2-75*x)*exp(x)+exp(4)**2-2*x**2*exp(4)+x**4+150*x**2)/(x*exp(x)**2+(2*x*ex
p(4)-2*x**3)*exp(x)+x*exp(4)**2-2*x**3*exp(4)+x**5),x)

[Out]

log(x) + 75/(-x**2 + exp(x) + exp(4))

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