3.59.3 \(\int \frac {-9 x-51 x^2+18 x^3+(-3 x-35 x^2+12 x^3) \log (-3+x)+(-6 x^2+2 x^3) \log ^2(-3+x)+(-216 x+71 x^2+(-144 x+48 x^2) \log (-3+x)+(-24 x+8 x^2) \log ^2(-3+x)) \log (x)+(-234+76 x+(-150+50 x) \log (-3+x)+(-24+8 x) \log ^2(-3+x)) \log ^2(x)}{-54 x^2+18 x^3+(-36 x^2+12 x^3) \log (-3+x)+(-6 x^2+2 x^3) \log ^2(-3+x)+(-216 x+72 x^2+(-144 x+48 x^2) \log (-3+x)+(-24 x+8 x^2) \log ^2(-3+x)) \log (x)+(-216+72 x+(-144+48 x) \log (-3+x)+(-24+8 x) \log ^2(-3+x)) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ x+\frac {x}{2 (3+\log (-3+x)) \left (2+\frac {x}{\log (x)}\right )} \]

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Rubi [F]  time = 4.52, 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 {-9 x-51 x^2+18 x^3+\left (-3 x-35 x^2+12 x^3\right ) \log (-3+x)+\left (-6 x^2+2 x^3\right ) \log ^2(-3+x)+\left (-216 x+71 x^2+\left (-144 x+48 x^2\right ) \log (-3+x)+\left (-24 x+8 x^2\right ) \log ^2(-3+x)\right ) \log (x)+\left (-234+76 x+(-150+50 x) \log (-3+x)+(-24+8 x) \log ^2(-3+x)\right ) \log ^2(x)}{-54 x^2+18 x^3+\left (-36 x^2+12 x^3\right ) \log (-3+x)+\left (-6 x^2+2 x^3\right ) \log ^2(-3+x)+\left (-216 x+72 x^2+\left (-144 x+48 x^2\right ) \log (-3+x)+\left (-24 x+8 x^2\right ) \log ^2(-3+x)\right ) \log (x)+\left (-216+72 x+(-144+48 x) \log (-3+x)+(-24+8 x) \log ^2(-3+x)\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9*x - 51*x^2 + 18*x^3 + (-3*x - 35*x^2 + 12*x^3)*Log[-3 + x] + (-6*x^2 + 2*x^3)*Log[-3 + x]^2 + (-216*x
+ 71*x^2 + (-144*x + 48*x^2)*Log[-3 + x] + (-24*x + 8*x^2)*Log[-3 + x]^2)*Log[x] + (-234 + 76*x + (-150 + 50*x
)*Log[-3 + x] + (-24 + 8*x)*Log[-3 + x]^2)*Log[x]^2)/(-54*x^2 + 18*x^3 + (-36*x^2 + 12*x^3)*Log[-3 + x] + (-6*
x^2 + 2*x^3)*Log[-3 + x]^2 + (-216*x + 72*x^2 + (-144*x + 48*x^2)*Log[-3 + x] + (-24*x + 8*x^2)*Log[-3 + x]^2)
*Log[x] + (-216 + 72*x + (-144 + 48*x)*Log[-3 + x] + (-24 + 8*x)*Log[-3 + x]^2)*Log[x]^2),x]

[Out]

x + 3/(4*(3 + Log[-3 + x])) - (3 - x)/(4*(3 + Log[-3 + x])) + Defer[Int][x/((3 + Log[-3 + x])*(x + 2*Log[x])^2
), x]/2 + Defer[Int][x^2/((3 + Log[-3 + x])*(x + 2*Log[x])^2), x]/4 + (3*Defer[Int][1/((3 + Log[-3 + x])^2*(x
+ 2*Log[x])), x])/4 + (9*Defer[Int][1/((-3 + x)*(3 + Log[-3 + x])^2*(x + 2*Log[x])), x])/4 - (5*Defer[Int][x/(
(3 + Log[-3 + x])^2*(x + 2*Log[x])), x])/4 - Defer[Int][(x*Log[-3 + x])/((3 + Log[-3 + x])^2*(x + 2*Log[x])),
x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 x \left (-3-17 x+6 x^2\right )-x (-216+71 x) \log (x)-(-234+76 x) \log ^2(x)-2 (-3+x) \log ^2(-3+x) (x+2 \log (x))^2-(-3+x) \log (-3+x) \left (x+12 x^2+48 x \log (x)+50 \log ^2(x)\right )}{2 (3-x) (3+\log (-3+x))^2 (x+2 \log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {-3 x \left (-3-17 x+6 x^2\right )-x (-216+71 x) \log (x)-(-234+76 x) \log ^2(x)-2 (-3+x) \log ^2(-3+x) (x+2 \log (x))^2-(-3+x) \log (-3+x) \left (x+12 x^2+48 x \log (x)+50 \log ^2(x)\right )}{(3-x) (3+\log (-3+x))^2 (x+2 \log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {-117+38 x-75 \log (-3+x)+25 x \log (-3+x)-12 \log ^2(-3+x)+4 x \log ^2(-3+x)}{2 (-3+x) (3+\log (-3+x))^2}+\frac {x (2+x)}{2 (3+\log (-3+x)) (x+2 \log (x))^2}-\frac {x (-18+5 x-6 \log (-3+x)+2 x \log (-3+x))}{2 (-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-117+38 x-75 \log (-3+x)+25 x \log (-3+x)-12 \log ^2(-3+x)+4 x \log ^2(-3+x)}{(-3+x) (3+\log (-3+x))^2} \, dx+\frac {1}{4} \int \frac {x (2+x)}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \int \frac {x (-18+5 x-6 \log (-3+x)+2 x \log (-3+x))}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=\frac {1}{4} \int \left (4-\frac {x}{(-3+x) (3+\log (-3+x))^2}+\frac {1}{3+\log (-3+x)}\right ) \, dx+\frac {1}{4} \int \left (\frac {2 x}{(3+\log (-3+x)) (x+2 \log (x))^2}+\frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2}\right ) \, dx-\frac {1}{4} \int \left (\frac {-18+5 x-6 \log (-3+x)+2 x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {3 (-18+5 x-6 \log (-3+x)+2 x \log (-3+x))}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx\\ &=x-\frac {1}{4} \int \frac {x}{(-3+x) (3+\log (-3+x))^2} \, dx+\frac {1}{4} \int \frac {1}{3+\log (-3+x)} \, dx+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \int \frac {-18+5 x-6 \log (-3+x)+2 x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {3}{4} \int \frac {-18+5 x-6 \log (-3+x)+2 x \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \int \left (-\frac {18}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {5 x}{(3+\log (-3+x))^2 (x+2 \log (x))}-\frac {6 \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {2 x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {3+x}{x (3+\log (x))^2} \, dx,x,-3+x\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{3+\log (x)} \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {3}{4} \int \left (-\frac {18}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}+\frac {5 x}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}-\frac {6 \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}+\frac {2 x \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx\\ &=x+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (-3+x)\right )-\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{(3+\log (x))^2}+\frac {3}{x (3+\log (x))^2}\right ) \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {3}{2} \int \frac {\log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{2} \int \frac {x \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {x}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {\log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {\text {Ei}(3+\log (-3+x))}{4 e^3}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{(3+\log (x))^2} \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{x (3+\log (x))^2} \, dx,x,-3+x\right )-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {3}{2} \int \frac {\log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{2} \int \left (\frac {\log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {3 \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx-\frac {15}{4} \int \left (\frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {3}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {\log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {\text {Ei}(3+\log (-3+x))}{4 e^3}-\frac {3-x}{4 (3+\log (-3+x))}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{3+\log (x)} \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,3+\log (-3+x)\right )-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {45}{4} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {\text {Ei}(3+\log (-3+x))}{4 e^3}+\frac {3}{4 (3+\log (-3+x))}-\frac {3-x}{4 (3+\log (-3+x))}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (-3+x)\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {45}{4} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {3}{4 (3+\log (-3+x))}-\frac {3-x}{4 (3+\log (-3+x))}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {45}{4} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 44, normalized size = 1.76 \begin {gather*} \frac {1}{2} \left (2 x+\frac {x}{2 (3+\log (-3+x))}-\frac {x^2}{2 (3+\log (-3+x)) (x+2 \log (x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x - 51*x^2 + 18*x^3 + (-3*x - 35*x^2 + 12*x^3)*Log[-3 + x] + (-6*x^2 + 2*x^3)*Log[-3 + x]^2 + (-
216*x + 71*x^2 + (-144*x + 48*x^2)*Log[-3 + x] + (-24*x + 8*x^2)*Log[-3 + x]^2)*Log[x] + (-234 + 76*x + (-150
+ 50*x)*Log[-3 + x] + (-24 + 8*x)*Log[-3 + x]^2)*Log[x]^2)/(-54*x^2 + 18*x^3 + (-36*x^2 + 12*x^3)*Log[-3 + x]
+ (-6*x^2 + 2*x^3)*Log[-3 + x]^2 + (-216*x + 72*x^2 + (-144*x + 48*x^2)*Log[-3 + x] + (-24*x + 8*x^2)*Log[-3 +
x]^2)*Log[x] + (-216 + 72*x + (-144 + 48*x)*Log[-3 + x] + (-24 + 8*x)*Log[-3 + x]^2)*Log[x]^2),x]

[Out]

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

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fricas [B]  time = 0.65, size = 53, normalized size = 2.12 \begin {gather*} \frac {2 \, x^{2} \log \left (x - 3\right ) + 6 \, x^{2} + {\left (4 \, x \log \left (x - 3\right ) + 13 \, x\right )} \log \relax (x)}{2 \, {\left (x \log \left (x - 3\right ) + 2 \, {\left (\log \left (x - 3\right ) + 3\right )} \log \relax (x) + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-24)*log(x-3)^2+(50*x-150)*log(x-3)+76*x-234)*log(x)^2+((8*x^2-24*x)*log(x-3)^2+(48*x^2-144*x)
*log(x-3)+71*x^2-216*x)*log(x)+(2*x^3-6*x^2)*log(x-3)^2+(12*x^3-35*x^2-3*x)*log(x-3)+18*x^3-51*x^2-9*x)/(((8*x
-24)*log(x-3)^2+(48*x-144)*log(x-3)+72*x-216)*log(x)^2+((8*x^2-24*x)*log(x-3)^2+(48*x^2-144*x)*log(x-3)+72*x^2
-216*x)*log(x)+(2*x^3-6*x^2)*log(x-3)^2+(12*x^3-36*x^2)*log(x-3)+18*x^3-54*x^2),x, algorithm="fricas")

[Out]

1/2*(2*x^2*log(x - 3) + 6*x^2 + (4*x*log(x - 3) + 13*x)*log(x))/(x*log(x - 3) + 2*(log(x - 3) + 3)*log(x) + 3*
x)

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giac [A]  time = 0.25, size = 31, normalized size = 1.24 \begin {gather*} x + \frac {x \log \relax (x)}{2 \, {\left (x \log \left (x - 3\right ) + 2 \, \log \left (x - 3\right ) \log \relax (x) + 3 \, x + 6 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-24)*log(x-3)^2+(50*x-150)*log(x-3)+76*x-234)*log(x)^2+((8*x^2-24*x)*log(x-3)^2+(48*x^2-144*x)
*log(x-3)+71*x^2-216*x)*log(x)+(2*x^3-6*x^2)*log(x-3)^2+(12*x^3-35*x^2-3*x)*log(x-3)+18*x^3-51*x^2-9*x)/(((8*x
-24)*log(x-3)^2+(48*x-144)*log(x-3)+72*x-216)*log(x)^2+((8*x^2-24*x)*log(x-3)^2+(48*x^2-144*x)*log(x-3)+72*x^2
-216*x)*log(x)+(2*x^3-6*x^2)*log(x-3)^2+(12*x^3-36*x^2)*log(x-3)+18*x^3-54*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 24, normalized size = 0.96

method	result	size
risch	\(x +\frac {x \ln \relax (x )}{2 \left (2 \ln \relax (x )+x \right ) \left (\ln \left (x -3\right )+3\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x-24)*ln(x-3)^2+(50*x-150)*ln(x-3)+76*x-234)*ln(x)^2+((8*x^2-24*x)*ln(x-3)^2+(48*x^2-144*x)*ln(x-3)+7
1*x^2-216*x)*ln(x)+(2*x^3-6*x^2)*ln(x-3)^2+(12*x^3-35*x^2-3*x)*ln(x-3)+18*x^3-51*x^2-9*x)/(((8*x-24)*ln(x-3)^2
+(48*x-144)*ln(x-3)+72*x-216)*ln(x)^2+((8*x^2-24*x)*ln(x-3)^2+(48*x^2-144*x)*ln(x-3)+72*x^2-216*x)*ln(x)+(2*x^
3-6*x^2)*ln(x-3)^2+(12*x^3-36*x^2)*ln(x-3)+18*x^3-54*x^2),x,method=_RETURNVERBOSE)

[Out]

x+1/2*x*ln(x)/(2*ln(x)+x)/(ln(x-3)+3)

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maxima [B]  time = 0.40, size = 49, normalized size = 1.96 \begin {gather*} \frac {6 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x \log \relax (x)\right )} \log \left (x - 3\right ) + 13 \, x \log \relax (x)}{2 \, {\left ({\left (x + 2 \, \log \relax (x)\right )} \log \left (x - 3\right ) + 3 \, x + 6 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-24)*log(x-3)^2+(50*x-150)*log(x-3)+76*x-234)*log(x)^2+((8*x^2-24*x)*log(x-3)^2+(48*x^2-144*x)
*log(x-3)+71*x^2-216*x)*log(x)+(2*x^3-6*x^2)*log(x-3)^2+(12*x^3-35*x^2-3*x)*log(x-3)+18*x^3-51*x^2-9*x)/(((8*x
-24)*log(x-3)^2+(48*x-144)*log(x-3)+72*x-216)*log(x)^2+((8*x^2-24*x)*log(x-3)^2+(48*x^2-144*x)*log(x-3)+72*x^2
-216*x)*log(x)+(2*x^3-6*x^2)*log(x-3)^2+(12*x^3-36*x^2)*log(x-3)+18*x^3-54*x^2),x, algorithm="maxima")

[Out]

1/2*(6*x^2 + 2*(x^2 + 2*x*log(x))*log(x - 3) + 13*x*log(x))/((x + 2*log(x))*log(x - 3) + 3*x + 6*log(x))

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mupad [B]  time = 4.38, size = 292, normalized size = 11.68 \begin {gather*} \frac {x}{2}-\ln \relax (x)+\frac {\frac {x\,\left (3\,x^4+14\,x^3-2\,x^2+12\,x+24\right )}{8\,{\left (x+2\right )}^3}+\frac {2\,x\,{\ln \relax (x)}^2\,\left (x^2+4\,x-3\right )}{{\left (x+2\right )}^3}-\frac {x\,\ln \relax (x)\,\left (-3\,x^3-13\,x^2+12\,x+6\right )}{2\,{\left (x+2\right )}^3}}{x+2\,\ln \relax (x)}+\frac {\frac {x^2\,\ln \relax (x)-3\,x^2-4\,x\,{\ln \relax (x)}^2+9\,x+18\,{\ln \relax (x)}^2}{2\,{\left (x+2\,\ln \relax (x)\right )}^2}-\frac {\ln \left (x-3\right )\,\left (2\,{\ln \relax (x)}^2+x\right )\,\left (x-3\right )}{2\,{\left (x+2\,\ln \relax (x)\right )}^2}}{\ln \left (x-3\right )+3}-\frac {\frac {x^2}{2}+\frac {27\,x}{2}+16}{x^3+6\,x^2+12\,x+8}+\frac {\frac {x\,\left (x^3+2\,x-24\right )}{8\,\left (x+2\right )}+\frac {x\,\ln \relax (x)\,\left (3\,x^2-8\,x+18\right )}{4\,\left (x+2\right )}+\frac {x\,{\ln \relax (x)}^2\,\left (2\,x-3\right )}{x+2}}{x^2+4\,x\,\ln \relax (x)+4\,{\ln \relax (x)}^2}+\frac {\ln \relax (x)\,\left (2\,x^2+15\,x+8\right )}{x^3+6\,x^2+12\,x+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x + log(x)*(216*x + log(x - 3)*(144*x - 48*x^2) + log(x - 3)^2*(24*x - 8*x^2) - 71*x^2) + log(x - 3)*(3
*x + 35*x^2 - 12*x^3) + log(x - 3)^2*(6*x^2 - 2*x^3) + 51*x^2 - 18*x^3 - log(x)^2*(76*x + log(x - 3)^2*(8*x -
24) + log(x - 3)*(50*x - 150) - 234))/(log(x)*(216*x + log(x - 3)*(144*x - 48*x^2) + log(x - 3)^2*(24*x - 8*x^
2) - 72*x^2) + log(x - 3)*(36*x^2 - 12*x^3) + log(x - 3)^2*(6*x^2 - 2*x^3) + 54*x^2 - 18*x^3 - log(x)^2*(72*x
+ log(x - 3)^2*(8*x - 24) + log(x - 3)*(48*x - 144) - 216)),x)

[Out]

x/2 - log(x) + ((x*(12*x - 2*x^2 + 14*x^3 + 3*x^4 + 24))/(8*(x + 2)^3) + (2*x*log(x)^2*(4*x + x^2 - 3))/(x + 2
)^3 - (x*log(x)*(12*x - 13*x^2 - 3*x^3 + 6))/(2*(x + 2)^3))/(x + 2*log(x)) + ((9*x - 4*x*log(x)^2 + x^2*log(x)
+ 18*log(x)^2 - 3*x^2)/(2*(x + 2*log(x))^2) - (log(x - 3)*(x + 2*log(x)^2)*(x - 3))/(2*(x + 2*log(x))^2))/(lo
g(x - 3) + 3) - ((27*x)/2 + x^2/2 + 16)/(12*x + 6*x^2 + x^3 + 8) + ((x*(2*x + x^3 - 24))/(8*(x + 2)) + (x*log(
x)*(3*x^2 - 8*x + 18))/(4*(x + 2)) + (x*log(x)^2*(2*x - 3))/(x + 2))/(4*log(x)^2 + 4*x*log(x) + x^2) + (log(x)
*(15*x + 2*x^2 + 8))/(12*x + 6*x^2 + x^3 + 8)

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sympy [A]  time = 0.36, size = 27, normalized size = 1.08 \begin {gather*} x + \frac {x \log {\relax (x )}}{6 x + \left (2 x + 4 \log {\relax (x )}\right ) \log {\left (x - 3 \right )} + 12 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-24)*ln(x-3)**2+(50*x-150)*ln(x-3)+76*x-234)*ln(x)**2+((8*x**2-24*x)*ln(x-3)**2+(48*x**2-144*x
)*ln(x-3)+71*x**2-216*x)*ln(x)+(2*x**3-6*x**2)*ln(x-3)**2+(12*x**3-35*x**2-3*x)*ln(x-3)+18*x**3-51*x**2-9*x)/(
((8*x-24)*ln(x-3)**2+(48*x-144)*ln(x-3)+72*x-216)*ln(x)**2+((8*x**2-24*x)*ln(x-3)**2+(48*x**2-144*x)*ln(x-3)+7
2*x**2-216*x)*ln(x)+(2*x**3-6*x**2)*ln(x-3)**2+(12*x**3-36*x**2)*ln(x-3)+18*x**3-54*x**2),x)

[Out]

x + x*log(x)/(6*x + (2*x + 4*log(x))*log(x - 3) + 12*log(x))

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