3.102.17 \(\int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 (7-12 x+3 x^2+2 x^3)+(26 x-22 x^2-6 x^3+e^6 (14-10 x-4 x^2)) \log (3)+(14 x+3 x^2+e^6 (7+2 x)) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 (7 x-13 x^2+5 x^3+x^4)+(-4+4 x+13 x^2-12 x^3-2 x^4+e^6 (14 x-12 x^2-2 x^3)) \log (3)+(-2+7 x^2+x^3+e^6 (7 x+x^2)) \log ^2(3)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (-2+x \left ((7+x) \left (e^6+x\right )+\frac {x}{-1+x-\log (3)}\right )\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.45, antiderivative size = 68, normalized size of antiderivative = 2.62, number of steps used = 3, number of rules used = 2, integrand size = 206, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2074, 1587} \begin {gather*} \log \left (x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+2+\log (9)\right )-\log (-x+1+\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12*x - 24*x^2 + 8*x^3 + 3*x^4 + E^6*(7 - 12*x + 3*x^2 + 2*x^3) + (26*x - 22*x^2 - 6*x^3 + E^6*(14 - 10*x
- 4*x^2))*Log[3] + (14*x + 3*x^2 + E^6*(7 + 2*x))*Log[3]^2)/(-2 + 4*x + 4*x^2 - 12*x^3 + 5*x^4 + x^5 + E^6*(7*
x - 13*x^2 + 5*x^3 + x^4) + (-4 + 4*x + 13*x^2 - 12*x^3 - 2*x^4 + E^6*(14*x - 12*x^2 - 2*x^3))*Log[3] + (-2 +
7*x^2 + x^3 + E^6*(7*x + x^2))*Log[3]^2),x]

[Out]

-Log[1 - x + Log[3]] + Log[2 + x^4 + x^3*(6 + E^6 - Log[3]) - x^2*(6 - E^6*(6 - Log[3]) + 7*Log[3]) - x*(2 + 7
*E^6*(1 + Log[3])) + Log[9]]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{1-x+\log (3)}+\frac {-2+4 x^3+3 x^2 \left (6+e^6-\log (3)\right )-7 e^6 (1+\log (3))-2 x \left (6-e^6 (6-\log (3))+7 \log (3)\right )}{2+x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+\log (9)}\right ) \, dx\\ &=-\log (1-x+\log (3))+\int \frac {-2+4 x^3+3 x^2 \left (6+e^6-\log (3)\right )-7 e^6 (1+\log (3))-2 x \left (6-e^6 (6-\log (3))+7 \log (3)\right )}{2+x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+\log (9)} \, dx\\ &=-\log (1-x+\log (3))+\log \left (2+x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+\log (9)\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [C]  time = 42.80, size = 588, normalized size = 22.62 \begin {gather*} \text {RootSum}\left [-2-4 \log (3)-2 \log ^2(3)+4 \text {$\#$1}+7 e^6 \text {$\#$1}+14 e^6 \log (3) \text {$\#$1}+7 e^6 \log ^2(3) \text {$\#$1}+\log (81) \text {$\#$1}+4 \text {$\#$1}^2-13 e^6 \text {$\#$1}^2+13 \log (3) \text {$\#$1}^2-12 e^6 \log (3) \text {$\#$1}^2+7 \log ^2(3) \text {$\#$1}^2+e^6 \log ^2(3) \text {$\#$1}^2-12 \text {$\#$1}^3+5 e^6 \text {$\#$1}^3-12 \log (3) \text {$\#$1}^3+\log ^2(3) \text {$\#$1}^3-e^6 \log (9) \text {$\#$1}^3+5 \text {$\#$1}^4+e^6 \text {$\#$1}^4-\log (9) \text {$\#$1}^4+\text {$\#$1}^5\&,\frac {7 e^6 \log (x-\text {$\#$1})+14 e^6 \log (3) \log (x-\text {$\#$1})+7 e^6 \log ^2(3) \log (x-\text {$\#$1})+12 \log (x-\text {$\#$1}) \text {$\#$1}-12 e^6 \log (x-\text {$\#$1}) \text {$\#$1}+26 \log (3) \log (x-\text {$\#$1}) \text {$\#$1}-10 e^6 \log (3) \log (x-\text {$\#$1}) \text {$\#$1}+14 \log ^2(3) \log (x-\text {$\#$1}) \text {$\#$1}+2 e^6 \log ^2(3) \log (x-\text {$\#$1}) \text {$\#$1}-24 \log (x-\text {$\#$1}) \text {$\#$1}^2+3 e^6 \log (x-\text {$\#$1}) \text {$\#$1}^2-22 \log (3) \log (x-\text {$\#$1}) \text {$\#$1}^2-4 e^6 \log (3) \log (x-\text {$\#$1}) \text {$\#$1}^2+3 \log ^2(3) \log (x-\text {$\#$1}) \text {$\#$1}^2+8 \log (x-\text {$\#$1}) \text {$\#$1}^3+2 e^6 \log (x-\text {$\#$1}) \text {$\#$1}^3-6 \log (3) \log (x-\text {$\#$1}) \text {$\#$1}^3+3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{4+7 e^6+14 e^6 \log (3)+7 e^6 \log ^2(3)+\log (81)+8 \text {$\#$1}-26 e^6 \text {$\#$1}+26 \log (3) \text {$\#$1}-24 e^6 \log (3) \text {$\#$1}+14 \log ^2(3) \text {$\#$1}+2 e^6 \log ^2(3) \text {$\#$1}-36 \text {$\#$1}^2+15 e^6 \text {$\#$1}^2-36 \log (3) \text {$\#$1}^2+3 \log ^2(3) \text {$\#$1}^2-3 e^6 \log (9) \text {$\#$1}^2+20 \text {$\#$1}^3+4 e^6 \text {$\#$1}^3-4 \log (9) \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*x - 24*x^2 + 8*x^3 + 3*x^4 + E^6*(7 - 12*x + 3*x^2 + 2*x^3) + (26*x - 22*x^2 - 6*x^3 + E^6*(14 -
10*x - 4*x^2))*Log[3] + (14*x + 3*x^2 + E^6*(7 + 2*x))*Log[3]^2)/(-2 + 4*x + 4*x^2 - 12*x^3 + 5*x^4 + x^5 + E
^6*(7*x - 13*x^2 + 5*x^3 + x^4) + (-4 + 4*x + 13*x^2 - 12*x^3 - 2*x^4 + E^6*(14*x - 12*x^2 - 2*x^3))*Log[3] +
(-2 + 7*x^2 + x^3 + E^6*(7*x + x^2))*Log[3]^2),x]

[Out]

RootSum[-2 - 4*Log[3] - 2*Log[3]^2 + 4*#1 + 7*E^6*#1 + 14*E^6*Log[3]*#1 + 7*E^6*Log[3]^2*#1 + Log[81]*#1 + 4*#
1^2 - 13*E^6*#1^2 + 13*Log[3]*#1^2 - 12*E^6*Log[3]*#1^2 + 7*Log[3]^2*#1^2 + E^6*Log[3]^2*#1^2 - 12*#1^3 + 5*E^
6*#1^3 - 12*Log[3]*#1^3 + Log[3]^2*#1^3 - E^6*Log[9]*#1^3 + 5*#1^4 + E^6*#1^4 - Log[9]*#1^4 + #1^5 & , (7*E^6*
Log[x - #1] + 14*E^6*Log[3]*Log[x - #1] + 7*E^6*Log[3]^2*Log[x - #1] + 12*Log[x - #1]*#1 - 12*E^6*Log[x - #1]*
#1 + 26*Log[3]*Log[x - #1]*#1 - 10*E^6*Log[3]*Log[x - #1]*#1 + 14*Log[3]^2*Log[x - #1]*#1 + 2*E^6*Log[3]^2*Log
[x - #1]*#1 - 24*Log[x - #1]*#1^2 + 3*E^6*Log[x - #1]*#1^2 - 22*Log[3]*Log[x - #1]*#1^2 - 4*E^6*Log[3]*Log[x -
#1]*#1^2 + 3*Log[3]^2*Log[x - #1]*#1^2 + 8*Log[x - #1]*#1^3 + 2*E^6*Log[x - #1]*#1^3 - 6*Log[3]*Log[x - #1]*#
1^3 + 3*Log[x - #1]*#1^4)/(4 + 7*E^6 + 14*E^6*Log[3] + 7*E^6*Log[3]^2 + Log[81] + 8*#1 - 26*E^6*#1 + 26*Log[3]
*#1 - 24*E^6*Log[3]*#1 + 14*Log[3]^2*#1 + 2*E^6*Log[3]^2*#1 - 36*#1^2 + 15*E^6*#1^2 - 36*Log[3]*#1^2 + 3*Log[3
]^2*#1^2 - 3*E^6*Log[9]*#1^2 + 20*#1^3 + 4*E^6*#1^3 - 4*Log[9]*#1^3 + 5*#1^4) & ]

________________________________________________________________________________________

fricas [B]  time = 0.63, size = 69, normalized size = 2.65 \begin {gather*} \log \left (x^{4} + 6 \, x^{3} - 6 \, x^{2} + {\left (x^{3} + 6 \, x^{2} - 7 \, x\right )} e^{6} - {\left (x^{3} + 7 \, x^{2} + {\left (x^{2} + 7 \, x\right )} e^{6} - 2\right )} \log \relax (3) - 2 \, x + 2\right ) - \log \left (x - \log \relax (3) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((7+2*x)*exp(3)^2+3*x^2+14*x)*log(3)^2+((-4*x^2-10*x+14)*exp(3)^2-6*x^3-22*x^2+26*x)*log(3)+(2*x^3+
3*x^2-12*x+7)*exp(3)^2+3*x^4+8*x^3-24*x^2+12*x)/(((x^2+7*x)*exp(3)^2+x^3+7*x^2-2)*log(3)^2+((-2*x^3-12*x^2+14*
x)*exp(3)^2-2*x^4-12*x^3+13*x^2+4*x-4)*log(3)+(x^4+5*x^3-13*x^2+7*x)*exp(3)^2+x^5+5*x^4-12*x^3+4*x^2+4*x-2),x,
algorithm="fricas")

[Out]

log(x^4 + 6*x^3 - 6*x^2 + (x^3 + 6*x^2 - 7*x)*e^6 - (x^3 + 7*x^2 + (x^2 + 7*x)*e^6 - 2)*log(3) - 2*x + 2) - lo
g(x - log(3) - 1)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((7+2*x)*exp(3)^2+3*x^2+14*x)*log(3)^2+((-4*x^2-10*x+14)*exp(3)^2-6*x^3-22*x^2+26*x)*log(3)+(2*x^3+
3*x^2-12*x+7)*exp(3)^2+3*x^4+8*x^3-24*x^2+12*x)/(((x^2+7*x)*exp(3)^2+x^3+7*x^2-2)*log(3)^2+((-2*x^3-12*x^2+14*
x)*exp(3)^2-2*x^4-12*x^3+13*x^2+4*x-4)*log(3)+(x^4+5*x^3-13*x^2+7*x)*exp(3)^2+x^5+5*x^4-12*x^3+4*x^2+4*x-2),x,
algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.31, size = 68, normalized size = 2.62

method	result	size
risch	\(-\ln \left (1+\ln \relax (3)-x \right )+\ln \left (x^{4}+\left (-\ln \relax (3)+{\mathrm e}^{6}+6\right ) x^{3}+\left (-{\mathrm e}^{6} \ln \relax (3)-7 \ln \relax (3)+6 \,{\mathrm e}^{6}-6\right ) x^{2}+\left (-7 \,{\mathrm e}^{6} \ln \relax (3)-7 \,{\mathrm e}^{6}-2\right ) x +2 \ln \relax (3)+2\right )\)	\(68\)
default	\(\ln \left (-\ln \relax (3) {\mathrm e}^{6} x^{2}+x^{3} {\mathrm e}^{6}-x^{3} \ln \relax (3)+x^{4}-7 x \,{\mathrm e}^{6} \ln \relax (3)+6 x^{2} {\mathrm e}^{6}-7 x^{2} \ln \relax (3)+6 x^{3}-7 x \,{\mathrm e}^{6}-6 x^{2}+2 \ln \relax (3)-2 x +2\right )-\ln \left (x -\ln \relax (3)-1\right )\)	\(83\)
norman	\(-\ln \left (1+\ln \relax (3)-x \right )+\ln \left (\ln \relax (3) {\mathrm e}^{6} x^{2}-x^{3} {\mathrm e}^{6}+x^{3} \ln \relax (3)-x^{4}+7 x \,{\mathrm e}^{6} \ln \relax (3)-6 x^{2} {\mathrm e}^{6}+7 x^{2} \ln \relax (3)-6 x^{3}+7 x \,{\mathrm e}^{6}+6 x^{2}-2 \ln \relax (3)+2 x -2\right )\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((7+2*x)*exp(3)^2+3*x^2+14*x)*ln(3)^2+((-4*x^2-10*x+14)*exp(3)^2-6*x^3-22*x^2+26*x)*ln(3)+(2*x^3+3*x^2-12
*x+7)*exp(3)^2+3*x^4+8*x^3-24*x^2+12*x)/(((x^2+7*x)*exp(3)^2+x^3+7*x^2-2)*ln(3)^2+((-2*x^3-12*x^2+14*x)*exp(3)
^2-2*x^4-12*x^3+13*x^2+4*x-4)*ln(3)+(x^4+5*x^3-13*x^2+7*x)*exp(3)^2+x^5+5*x^4-12*x^3+4*x^2+4*x-2),x,method=_RE
TURNVERBOSE)

[Out]

-ln(1+ln(3)-x)+ln(x^4+(-ln(3)+exp(6)+6)*x^3+(-exp(6)*ln(3)-7*ln(3)+6*exp(6)-6)*x^2+(-7*exp(6)*ln(3)-7*exp(6)-2
)*x+2*ln(3)+2)

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 66, normalized size = 2.54 \begin {gather*} \log \left (x^{4} + x^{3} {\left (e^{6} - \log \relax (3) + 6\right )} - {\left ({\left (e^{6} + 7\right )} \log \relax (3) - 6 \, e^{6} + 6\right )} x^{2} - {\left (7 \, e^{6} \log \relax (3) + 7 \, e^{6} + 2\right )} x + 2 \, \log \relax (3) + 2\right ) - \log \left (x - \log \relax (3) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((7+2*x)*exp(3)^2+3*x^2+14*x)*log(3)^2+((-4*x^2-10*x+14)*exp(3)^2-6*x^3-22*x^2+26*x)*log(3)+(2*x^3+
3*x^2-12*x+7)*exp(3)^2+3*x^4+8*x^3-24*x^2+12*x)/(((x^2+7*x)*exp(3)^2+x^3+7*x^2-2)*log(3)^2+((-2*x^3-12*x^2+14*
x)*exp(3)^2-2*x^4-12*x^3+13*x^2+4*x-4)*log(3)+(x^4+5*x^3-13*x^2+7*x)*exp(3)^2+x^5+5*x^4-12*x^3+4*x^2+4*x-2),x,
algorithm="maxima")

[Out]

log(x^4 + x^3*(e^6 - log(3) + 6) - ((e^6 + 7)*log(3) - 6*e^6 + 6)*x^2 - (7*e^6*log(3) + 7*e^6 + 2)*x + 2*log(3
) + 2) - log(x - log(3) - 1)

________________________________________________________________________________________

mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + exp(6)*(3*x^2 - 12*x + 2*x^3 + 7) + log(3)^2*(14*x + 3*x^2 + exp(6)*(2*x + 7)) - log(3)*(exp(6)*(1
0*x + 4*x^2 - 14) - 26*x + 22*x^2 + 6*x^3) - 24*x^2 + 8*x^3 + 3*x^4)/(4*x - log(3)*(exp(6)*(12*x^2 - 14*x + 2*
x^3) - 4*x - 13*x^2 + 12*x^3 + 2*x^4 + 4) + exp(6)*(7*x - 13*x^2 + 5*x^3 + x^4) + log(3)^2*(exp(6)*(7*x + x^2)
+ 7*x^2 + x^3 - 2) + 4*x^2 - 12*x^3 + 5*x^4 + x^5 - 2),x)

[Out]

\text{Hanged}

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((7+2*x)*exp(3)**2+3*x**2+14*x)*ln(3)**2+((-4*x**2-10*x+14)*exp(3)**2-6*x**3-22*x**2+26*x)*ln(3)+(2
*x**3+3*x**2-12*x+7)*exp(3)**2+3*x**4+8*x**3-24*x**2+12*x)/(((x**2+7*x)*exp(3)**2+x**3+7*x**2-2)*ln(3)**2+((-2
*x**3-12*x**2+14*x)*exp(3)**2-2*x**4-12*x**3+13*x**2+4*x-4)*ln(3)+(x**4+5*x**3-13*x**2+7*x)*exp(3)**2+x**5+5*x
**4-12*x**3+4*x**2+4*x-2),x)

[Out]

Timed out

________________________________________________________________________________________