3.55.67 \(\int \frac {-x^2+e^6 (9+16 x-x^2)+(x^2+e^6 (-3-8 x+x^2)) \log (-\frac {2 e^6 x^3}{x^2+e^6 (-3-8 x+x^2)})}{(x^2+e^6 (-3-8 x+x^2)) \log ^2(-\frac {2 e^6 x^3}{x^2+e^6 (-3-8 x+x^2)})} \, dx\)

Optimal. Leaf size=32 \[ \frac {x}{\log \left (\frac {x^2}{4+\frac {3-x \left (x+\frac {x}{e^6}\right )}{2 x}}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 2.15, 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 {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + E^6*(9 + 16*x - x^2) + (x^2 + E^6*(-3 - 8*x + x^2))*Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))]
)/((x^2 + E^6*(-3 - 8*x + x^2))*Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))]^2),x]

[Out]

-Defer[Int][Log[(-2*E^6*x^3)/(-3*E^6 - 8*E^6*x + (1 + E^6)*x^2)]^(-2), x] + (6*E^3*(1 + E^6)*Defer[Int][1/((8*
E^6 - 2*E^3*Sqrt[3 + 19*E^6] - 2*(1 + E^6)*x)*Log[(-2*E^6*x^3)/(-3*E^6 - 8*E^6*x + (1 + E^6)*x^2)]^2), x])/Sqr
t[3 + 19*E^6] + (6*E^3*(1 + E^6)*Defer[Int][1/((-8*E^6 - 2*E^3*Sqrt[3 + 19*E^6] + 2*(1 + E^6)*x)*Log[(-2*E^6*x
^3)/(-3*E^6 - 8*E^6*x + (1 + E^6)*x^2)]^2), x])/Sqrt[3 + 19*E^6] + 8*E^6*(1 + (4*E^3)/Sqrt[3 + 19*E^6])*Defer[
Int][1/((-8*E^6 - 2*E^3*Sqrt[3 + 19*E^6] + 2*(1 + E^6)*x)*Log[(-2*E^6*x^3)/(-3*E^6 - 8*E^6*x + (1 + E^6)*x^2)]
^2), x] + 8*E^6*(1 - (4*E^3)/Sqrt[3 + 19*E^6])*Defer[Int][1/((-8*E^6 + 2*E^3*Sqrt[3 + 19*E^6] + 2*(1 + E^6)*x)
*Log[(-2*E^6*x^3)/(-3*E^6 - 8*E^6*x + (1 + E^6)*x^2)]^2), x] + Defer[Int][Log[(-2*E^6*x^3)/(-3*E^6 - 8*E^6*x +
(1 + E^6)*x^2)]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2-e^6 \left (9+16 x-x^2\right )-\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\int \left (\frac {-9 e^6-16 e^6 x+\left (1+e^6\right ) x^2}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx\\ &=\int \frac {-9 e^6-16 e^6 x+\left (1+e^6\right ) x^2}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\int \left (-\frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {2 e^6 (-3-4 x)}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (2 e^6\right ) \int \frac {-3-4 x}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (2 e^6\right ) \int \left (\frac {3}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {4 x}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (6 e^6\right ) \int \frac {1}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\left (8 e^6\right ) \int \frac {x}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (6 e^6\right ) \int \left (\frac {1+e^6}{e^3 \sqrt {3+19 e^6} \left (8 e^6-2 e^3 \sqrt {3+19 e^6}-2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {1+e^6}{e^3 \sqrt {3+19 e^6} \left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx+\left (8 e^6\right ) \int \left (\frac {1+\frac {4 e^3}{\sqrt {3+19 e^6}}}{\left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {1-\frac {4 e^3}{\sqrt {3+19 e^6}}}{\left (-8 e^6+2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\frac {\left (6 e^3 \left (1+e^6\right )\right ) \int \frac {1}{\left (8 e^6-2 e^3 \sqrt {3+19 e^6}-2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx}{\sqrt {3+19 e^6}}+\frac {\left (6 e^3 \left (1+e^6\right )\right ) \int \frac {1}{\left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx}{\sqrt {3+19 e^6}}+\left (8 e^6 \left (1-\frac {4 e^3}{\sqrt {3+19 e^6}}\right )\right ) \int \frac {1}{\left (-8 e^6+2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\left (8 e^6 \left (1+\frac {4 e^3}{\sqrt {3+19 e^6}}\right )\right ) \int \frac {1}{\left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 31, normalized size = 0.97 \begin {gather*} \frac {x}{\log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + E^6*(9 + 16*x - x^2) + (x^2 + E^6*(-3 - 8*x + x^2))*Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x +
x^2))])/((x^2 + E^6*(-3 - 8*x + x^2))*Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))]^2),x]

[Out]

x/Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))]

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 29, normalized size = 0.91 \begin {gather*} \frac {x}{\log \left (-\frac {2 \, x^{3} e^{6}}{x^{2} + {\left (x^{2} - 8 \, x - 3\right )} e^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x
^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x, algorithm="fricas")

[Out]

x/log(-2*x^3*e^6/(x^2 + (x^2 - 8*x - 3)*e^6))

________________________________________________________________________________________

giac [A]  time = 0.30, size = 33, normalized size = 1.03 \begin {gather*} \frac {x}{\log \left (-\frac {2 \, x^{3}}{x^{2} e^{6} + x^{2} - 8 \, x e^{6} - 3 \, e^{6}}\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x
^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x, algorithm="giac")

[Out]

x/(log(-2*x^3/(x^2*e^6 + x^2 - 8*x*e^6 - 3*e^6)) + 6)

________________________________________________________________________________________

maple [A]  time = 0.42, size = 30, normalized size = 0.94

method	result	size
risch	\(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{\left (x^{2}-8 x -3\right ) {\mathrm e}^{6}+x^{2}}\right )}\)	\(30\)
norman	\(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{\left (x^{2}-8 x -3\right ) {\mathrm e}^{6}+x^{2}}\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-8*x-3)*exp(3)^2+x^2)*ln(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x^2)/((x
^2-8*x-3)*exp(3)^2+x^2)/ln(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x,method=_RETURNVERBOSE)

[Out]

x/ln(-2*x^3*exp(6)/((x^2-8*x-3)*exp(6)+x^2))

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 34, normalized size = 1.06 \begin {gather*} \frac {x}{\log \relax (2) - \log \left (-x^{2} {\left (e^{6} + 1\right )} + 8 \, x e^{6} + 3 \, e^{6}\right ) + 3 \, \log \relax (x) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x
^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x, algorithm="maxima")

[Out]

x/(log(2) - log(-x^2*(e^6 + 1) + 8*x*e^6 + 3*e^6) + 3*log(x) + 6)

________________________________________________________________________________________

mupad [B]  time = 5.63, size = 77, normalized size = 2.41 \begin {gather*} \frac {x+x\,{\mathrm {e}}^6-8\,\ln \left (\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^6\,\left (-x^2+8\,x+3\right )-x^2}\right )\,{\mathrm {e}}^6}{\ln \left (\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^6\,\left (-x^2+8\,x+3\right )-x^2}\right )\,\left ({\mathrm {e}}^6+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((2*x^3*exp(6))/(exp(6)*(8*x - x^2 + 3) - x^2))*(exp(6)*(8*x - x^2 + 3) - x^2) - exp(6)*(16*x - x^2 +
9) + x^2)/(log((2*x^3*exp(6))/(exp(6)*(8*x - x^2 + 3) - x^2))^2*(exp(6)*(8*x - x^2 + 3) - x^2)),x)

[Out]

(x + x*exp(6) - 8*log((2*x^3*exp(6))/(exp(6)*(8*x - x^2 + 3) - x^2))*exp(6))/(log((2*x^3*exp(6))/(exp(6)*(8*x
- x^2 + 3) - x^2))*(exp(6) + 1))

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 27, normalized size = 0.84 \begin {gather*} \frac {x}{\log {\left (- \frac {2 x^{3} e^{6}}{x^{2} + \left (x^{2} - 8 x - 3\right ) e^{6}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-8*x-3)*exp(3)**2+x**2)*ln(-2*x**3*exp(3)**2/((x**2-8*x-3)*exp(3)**2+x**2))+(-x**2+16*x+9)*ex
p(3)**2-x**2)/((x**2-8*x-3)*exp(3)**2+x**2)/ln(-2*x**3*exp(3)**2/((x**2-8*x-3)*exp(3)**2+x**2))**2,x)

[Out]

x/log(-2*x**3*exp(6)/(x**2 + (x**2 - 8*x - 3)*exp(6)))

________________________________________________________________________________________