3.35.23 \(\int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 (12-15 x-16 x^2+15 x^3)+(2 x^3-2 x^4+e^3 (-2 x^2+2 x^3)) \log (-x^5+x^6+e^3 (x^4-x^5))}{100 x^2-100 x^3+e^3 (-100 x+100 x^2)+(20 x^2-20 x^3+e^3 (-20 x+20 x^2)) \log (-x^5+x^6+e^3 (x^4-x^5))+(x^2-x^3+e^3 (-x+x^2)) \log ^2(-x^5+x^6+e^3 (x^4-x^5))} \, dx\)

Optimal. Leaf size=27 \[ \frac {3+x^2}{10+\log \left ((1-x) \left (e^3-x\right ) x^4\right )} \]

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Rubi [F]  time = 5.53, 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 {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15*x + 18*x^2 + 15*x^3 - 14*x^4 + E^3*(12 - 15*x - 16*x^2 + 15*x^3) + (2*x^3 - 2*x^4 + E^3*(-2*x^2 + 2*x
^3))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)])/(100*x^2 - 100*x^3 + E^3*(-100*x + 100*x^2) + (20*x^2 - 20*x^3 + E^3*(
-20*x + 20*x^2))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)] + (x^2 - x^3 + E^3*(-x + x^2))*Log[-x^5 + x^6 + E^3*(x^4 -
x^5)]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x \left (-15+18 x+15 x^2-14 x^3\right )-e^3 \left (12-15 x-16 x^2+15 x^3\right )-2 \left (e^3-x\right ) (-1+x) x^2 \log \left ((-1+x) x^4 \left (-e^3+x\right )\right )}{(1-x) \left (e^3-x\right ) x \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx\\ &=\int \left (\frac {\left (-4 e^3+5 \left (1+e^3\right ) x-6 x^2\right ) \left (3+x^2\right )}{(1-x) \left (e^3-x\right ) x \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2}+\frac {2 x}{10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )}\right ) \, dx\\ &=2 \int \frac {x}{10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )} \, dx+\int \frac {\left (-4 e^3+5 \left (1+e^3\right ) x-6 x^2\right ) \left (3+x^2\right )}{(1-x) \left (e^3-x\right ) x \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx\\ &=2 \int \frac {x}{10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )} \, dx+\int \left (-\frac {1+e^3}{\left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2}+\frac {3+e^6}{\left (e^3-x\right ) \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2}-\frac {4}{(-1+x) \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2}-\frac {12}{x \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2}-\frac {6 x}{\left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x}{10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )} \, dx-4 \int \frac {1}{(-1+x) \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx-6 \int \frac {x}{\left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx-12 \int \frac {1}{x \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx+\left (-1-e^3\right ) \int \frac {1}{\left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx+\left (3+e^6\right ) \int \frac {1}{\left (e^3-x\right ) \left (10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 25, normalized size = 0.93 \begin {gather*} \frac {3+x^2}{10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15*x + 18*x^2 + 15*x^3 - 14*x^4 + E^3*(12 - 15*x - 16*x^2 + 15*x^3) + (2*x^3 - 2*x^4 + E^3*(-2*x^2
+ 2*x^3))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)])/(100*x^2 - 100*x^3 + E^3*(-100*x + 100*x^2) + (20*x^2 - 20*x^3 +
E^3*(-20*x + 20*x^2))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)] + (x^2 - x^3 + E^3*(-x + x^2))*Log[-x^5 + x^6 + E^3*(
x^4 - x^5)]^2),x]

[Out]

(3 + x^2)/(10 + Log[(-1 + x)*x^4*(-E^3 + x)])

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fricas [A]  time = 0.60, size = 33, normalized size = 1.22 \begin {gather*} \frac {x^{2} + 3}{\log \left (x^{6} - x^{5} - {\left (x^{5} - x^{4}\right )} e^{3}\right ) + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14
*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp(3)-x^3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x
^3+20*x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100*x^2),x, algorithm="fricas")

[Out]

(x^2 + 3)/(log(x^6 - x^5 - (x^5 - x^4)*e^3) + 10)

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giac [A]  time = 0.49, size = 33, normalized size = 1.22 \begin {gather*} \frac {x^{2} + 3}{\log \left (x^{6} - x^{5} e^{3} - x^{5} + x^{4} e^{3}\right ) + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14
*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp(3)-x^3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x
^3+20*x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100*x^2),x, algorithm="giac")

[Out]

(x^2 + 3)/(log(x^6 - x^5*e^3 - x^5 + x^4*e^3) + 10)

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maple [A]  time = 0.28, size = 33, normalized size = 1.22

method	result	size
norman	\(\frac {x^{2}+3}{\ln \left (\left (-x^{5}+x^{4}\right ) {\mathrm e}^{3}+x^{6}-x^{5}\right )+10}\)	\(33\)
risch	\(\frac {x^{2}+3}{\ln \left (\left (-x^{5}+x^{4}\right ) {\mathrm e}^{3}+x^{6}-x^{5}\right )+10}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*ln((-x^5+x^4)*exp(3)+x^6-x^5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14*x^4+15
*x^3+18*x^2-15*x)/(((x^2-x)*exp(3)-x^3+x^2)*ln((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x^3+20*x^
2)*ln((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100*x^2),x,method=_RETURNVERBOSE)

[Out]

(x^2+3)/(ln((-x^5+x^4)*exp(3)+x^6-x^5)+10)

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maxima [A]  time = 0.45, size = 25, normalized size = 0.93 \begin {gather*} \frac {x^{2} + 3}{\log \left (x - e^{3}\right ) + \log \left (x - 1\right ) + 4 \, \log \relax (x) + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14
*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp(3)-x^3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x
^3+20*x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100*x^2),x, algorithm="maxima")

[Out]

(x^2 + 3)/(log(x - e^3) + log(x - 1) + 4*log(x) + 10)

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mupad [B]  time = 11.35, size = 213, normalized size = 7.89 \begin {gather*} \frac {\frac {10\,{\mathrm {e}}^3}{3}-\frac {4\,{\mathrm {e}}^6}{3}+\frac {10\,{\mathrm {e}}^9}{3}+x\,\left (\frac {3\,{\mathrm {e}}^3}{2}+\frac {3\,{\mathrm {e}}^6}{2}-\frac {25\,{\mathrm {e}}^9}{6}-\frac {25}{6}\right )}{108\,x^2+\left (-90\,{\mathrm {e}}^3-90\right )\,x+72\,{\mathrm {e}}^3}+\frac {\frac {15\,x-12\,{\mathrm {e}}^3+15\,x\,{\mathrm {e}}^3+16\,x^2\,{\mathrm {e}}^3-15\,x^3\,{\mathrm {e}}^3-18\,x^2-15\,x^3+14\,x^4}{5\,x-4\,{\mathrm {e}}^3+5\,x\,{\mathrm {e}}^3-6\,x^2}+\frac {2\,x^2\,\ln \left ({\mathrm {e}}^3\,\left (x^4-x^5\right )-x^5+x^6\right )\,\left (x-{\mathrm {e}}^3\right )\,\left (x-1\right )}{5\,x-4\,{\mathrm {e}}^3+5\,x\,{\mathrm {e}}^3-6\,x^2}}{\ln \left ({\mathrm {e}}^3\,\left (x^4-x^5\right )-x^5+x^6\right )+10}+\frac {x^2}{3}-x\,\left (\frac {{\mathrm {e}}^3}{18}+\frac {1}{18}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*x + exp(3)*(15*x + 16*x^2 - 15*x^3 - 12) + log(exp(3)*(x^4 - x^5) - x^5 + x^6)*(exp(3)*(2*x^2 - 2*x^3)
- 2*x^3 + 2*x^4) - 18*x^2 - 15*x^3 + 14*x^4)/(log(exp(3)*(x^4 - x^5) - x^5 + x^6)^2*(exp(3)*(x - x^2) - x^2 +
x^3) + exp(3)*(100*x - 100*x^2) + log(exp(3)*(x^4 - x^5) - x^5 + x^6)*(exp(3)*(20*x - 20*x^2) - 20*x^2 + 20*x
^3) - 100*x^2 + 100*x^3),x)

[Out]

((10*exp(3))/3 - (4*exp(6))/3 + (10*exp(9))/3 + x*((3*exp(3))/2 + (3*exp(6))/2 - (25*exp(9))/6 - 25/6))/(72*ex
p(3) + 108*x^2 - x*(90*exp(3) + 90)) + ((15*x - 12*exp(3) + 15*x*exp(3) + 16*x^2*exp(3) - 15*x^3*exp(3) - 18*x
^2 - 15*x^3 + 14*x^4)/(5*x - 4*exp(3) + 5*x*exp(3) - 6*x^2) + (2*x^2*log(exp(3)*(x^4 - x^5) - x^5 + x^6)*(x -
exp(3))*(x - 1))/(5*x - 4*exp(3) + 5*x*exp(3) - 6*x^2))/(log(exp(3)*(x^4 - x^5) - x^5 + x^6) + 10) + x^2/3 - x
*(exp(3)/18 + 1/18)

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sympy [A]  time = 0.27, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{2} + 3}{\log {\left (x^{6} - x^{5} + \left (- x^{5} + x^{4}\right ) e^{3} \right )} + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3-2*x**2)*exp(3)-2*x**4+2*x**3)*ln((-x**5+x**4)*exp(3)+x**6-x**5)+(15*x**3-16*x**2-15*x+12)*
exp(3)-14*x**4+15*x**3+18*x**2-15*x)/(((x**2-x)*exp(3)-x**3+x**2)*ln((-x**5+x**4)*exp(3)+x**6-x**5)**2+((20*x*
*2-20*x)*exp(3)-20*x**3+20*x**2)*ln((-x**5+x**4)*exp(3)+x**6-x**5)+(100*x**2-100*x)*exp(3)-100*x**3+100*x**2),
x)

[Out]

(x**2 + 3)/(log(x**6 - x**5 + (-x**5 + x**4)*exp(3)) + 10)

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