∫ −507x2+481x3−152x4+16x5+(−234x2+150x3−24x4) log(81−108x+54x2−12x3+x4)+(−27x2+9x3) log2(81−108x+54x2−12x3+x4)+e 1 __________________________________________________________ 13x−4x2+3x log(81−108x+54x2−12x3+x4) (39−49x+8x2+(9−3x) log(81−108x+54x2−12x3+x4)) ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________−507x2 +481x3−152x4+16x5+(−234x2+150x3−24x4) log(81−108x+54x2−12x3+x4)+(−27x2+9x3) log2(81−108x+54x2−12x3+x4) dx

3.103.63 \(\int \frac {-507 x^2+481 x^3-152 x^4+16 x^5+(-234 x^2+150 x^3-24 x^4) \log (81-108 x+54 x^2-12 x^3+x^4)+(-27 x^2+9 x^3) \log ^2(81-108 x+54 x^2-12 x^3+x^4)+e^{\frac {1}{13 x-4 x^2+3 x \log (81-108 x+54 x^2-12 x^3+x^4)}} (39-49 x+8 x^2+(9-3 x) \log (81-108 x+54 x^2-12 x^3+x^4))}{-507 x^2+481 x^3-152 x^4+16 x^5+(-234 x^2+150 x^3-24 x^4) \log (81-108 x+54 x^2-12 x^3+x^4)+(-27 x^2+9 x^3) \log ^2(81-108 x+54 x^2-12 x^3+x^4)} \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {1}{x \left (4-4 x+3 \left (3+\log \left ((-3+x)^4\right )\right )\right )}}+x \]

________________________________________________________________________________________

Rubi [A] time = 3.93, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, integrand size = 254, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6688, 6742, 6706} \begin {gather*} x+e^{\frac {1}{x \left (-4 x+3 \log \left ((x-3)^4\right )+13\right )}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(-507*x^2 + 481*x^3 - 152*x^4 + 16*x^5 + (-234*x^2 + 150*x^3 - 24*x^4)*Log[81 - 108*x + 54*x^2 - 12*x^3 +

x^4] + (-27*x^2 + 9*x^3)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4]^2 + E^(13*x - 4*x^2 + 3*x*Log[81 - 108*x + 54

*x^2 - 12*x^3 + x^4])^(-1)*(39 - 49*x + 8*x^2 + (9 - 3*x)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4]))/(-507*x^2

+ 481*x^3 - 152*x^4 + 16*x^5 + (-234*x^2 + 150*x^3 - 24*x^4)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4] + (-27*x^

2 + 9*x^3)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4]^2),x]

[Out]

E^(1/(x*(13 - 4*x + 3*Log[(-3 + x)^4]))) + x

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /; !FalseQ[q]

] /; FreeQ[F, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-(13-4 x)^2 (-3+x) x^2-e^{-\frac {1}{x \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )}} \left (39-49 x+8 x^2\right )+3 e^{-\frac {1}{x \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )}} (-3+x) \left (1+2 e^{\frac {1}{x \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )}} x^2 (-13+4 x)\right ) \log \left ((-3+x)^4\right )-9 (-3+x) x^2 \log ^2\left ((-3+x)^4\right )}{(3-x) x^2 \left (13-4 x+3 \log \left ((-3+x)^4\right )\right )^2} \, dx\\ &=\int \left (1+\frac {e^{-\frac {1}{x \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )}} \left (39-49 x+8 x^2+9 \log \left ((-3+x)^4\right )-3 x \log \left ((-3+x)^4\right )\right )}{(-3+x) x^2 \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )^2}\right ) \, dx\\ &=x+\int \frac {e^{-\frac {1}{x \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )}} \left (39-49 x+8 x^2+9 \log \left ((-3+x)^4\right )-3 x \log \left ((-3+x)^4\right )\right )}{(-3+x) x^2 \left (-13+4 x-3 \log \left ((-3+x)^4\right )\right )^2} \, dx\\ &=e^{\frac {1}{x \left (13-4 x+3 \log \left ((-3+x)^4\right )\right )}}+x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.17, size = 23, normalized size = 0.92 \begin {gather*} e^{\frac {1}{x \left (13-4 x+3 \log \left ((-3+x)^4\right )\right )}}+x \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-507*x^2 + 481*x^3 - 152*x^4 + 16*x^5 + (-234*x^2 + 150*x^3 - 24*x^4)*Log[81 - 108*x + 54*x^2 - 12*

x^3 + x^4] + (-27*x^2 + 9*x^3)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4]^2 + E^(13*x - 4*x^2 + 3*x*Log[81 - 108*

x + 54*x^2 - 12*x^3 + x^4])^(-1)*(39 - 49*x + 8*x^2 + (9 - 3*x)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4]))/(-50

7*x^2 + 481*x^3 - 152*x^4 + 16*x^5 + (-234*x^2 + 150*x^3 - 24*x^4)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4] + (

-27*x^2 + 9*x^3)*Log[81 - 108*x + 54*x^2 - 12*x^3 + x^4]^2),x]

[Out]

E^(1/(x*(13 - 4*x + 3*Log[(-3 + x)^4]))) + x

________________________________________________________________________________________

fricas [A] time = 0.77, size = 38, normalized size = 1.52 \begin {gather*} x + e^{\left (-\frac {1}{4 \, x^{2} - 3 \, x \log \left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right ) - 13 \, x}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+9)*log(x^4-12*x^3+54*x^2-108*x+81)+8*x^2-49*x+39)*exp(1/(3*x*log(x^4-12*x^3+54*x^2-108*x+81)

-4*x^2+13*x))+(9*x^3-27*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-234*x^2)*log(x^4-12*x^3+54*x^2

-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2)/((9*x^3-27*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-

234*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2),x, algorithm="fricas")

[Out]

x + e^(-1/(4*x^2 - 3*x*log(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) - 13*x))

________________________________________________________________________________________

giac [A] time = 3.24, size = 38, normalized size = 1.52 \begin {gather*} x + e^{\left (-\frac {1}{4 \, x^{2} - 3 \, x \log \left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right ) - 13 \, x}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+9)*log(x^4-12*x^3+54*x^2-108*x+81)+8*x^2-49*x+39)*exp(1/(3*x*log(x^4-12*x^3+54*x^2-108*x+81)

-4*x^2+13*x))+(9*x^3-27*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-234*x^2)*log(x^4-12*x^3+54*x^2

-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2)/((9*x^3-27*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-

234*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2),x, algorithm="giac")

[Out]

x + e^(-1/(4*x^2 - 3*x*log(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) - 13*x))

________________________________________________________________________________________

maple [A] time = 0.25, size = 37, normalized size = 1.48


method	result	size

risch	\(x +{\mathrm e}^{-\frac {1}{x \left (-3 \ln \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )+4 x -13\right )}}\)	\(37\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x+9)*ln(x^4-12*x^3+54*x^2-108*x+81)+8*x^2-49*x+39)*exp(1/(3*x*ln(x^4-12*x^3+54*x^2-108*x+81)-4*x^2+1

3*x))+(9*x^3-27*x^2)*ln(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-234*x^2)*ln(x^4-12*x^3+54*x^2-108*x+81)

+16*x^5-152*x^4+481*x^3-507*x^2)/((9*x^3-27*x^2)*ln(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-234*x^2)*ln

(x^4-12*x^3+54*x^2-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2),x,method=_RETURNVERBOSE)

[Out]

x+exp(-1/x/(-3*ln(x^4-12*x^3+54*x^2-108*x+81)+4*x-13))

________________________________________________________________________________________

maxima [B] time = 0.43, size = 49, normalized size = 1.96 \begin {gather*} x + e^{\left (-\frac {4}{4 \, x {\left (12 \, \log \left (x - 3\right ) + 13\right )} - 144 \, \log \left (x - 3\right )^{2} - 312 \, \log \left (x - 3\right ) - 169} + \frac {1}{x {\left (12 \, \log \left (x - 3\right ) + 13\right )}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+9)*log(x^4-12*x^3+54*x^2-108*x+81)+8*x^2-49*x+39)*exp(1/(3*x*log(x^4-12*x^3+54*x^2-108*x+81)

-4*x^2+13*x))+(9*x^3-27*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-234*x^2)*log(x^4-12*x^3+54*x^2

-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2)/((9*x^3-27*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)^2+(-24*x^4+150*x^3-

234*x^2)*log(x^4-12*x^3+54*x^2-108*x+81)+16*x^5-152*x^4+481*x^3-507*x^2),x, algorithm="maxima")

[Out]

x + e^(-4/(4*x*(12*log(x - 3) + 13) - 144*log(x - 3)^2 - 312*log(x - 3) - 169) + 1/(x*(12*log(x - 3) + 13)))

________________________________________________________________________________________

mupad [B] time = 7.24, size = 36, normalized size = 1.44 \begin {gather*} x+{\mathrm {e}}^{\frac {1}{13\,x+3\,x\,\ln \left (x^4-12\,x^3+54\,x^2-108\,x+81\right )-4\,x^2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/(13*x + 3*x*log(54*x^2 - 108*x - 12*x^3 + x^4 + 81) - 4*x^2))*(49*x + log(54*x^2 - 108*x - 12*x^3 +

x^4 + 81)*(3*x - 9) - 8*x^2 - 39) + log(54*x^2 - 108*x - 12*x^3 + x^4 + 81)*(234*x^2 - 150*x^3 + 24*x^4) + lo

g(54*x^2 - 108*x - 12*x^3 + x^4 + 81)^2*(27*x^2 - 9*x^3) + 507*x^2 - 481*x^3 + 152*x^4 - 16*x^5)/(log(54*x^2 -

108*x - 12*x^3 + x^4 + 81)*(234*x^2 - 150*x^3 + 24*x^4) + log(54*x^2 - 108*x - 12*x^3 + x^4 + 81)^2*(27*x^2 -

9*x^3) + 507*x^2 - 481*x^3 + 152*x^4 - 16*x^5),x)

[Out]

x + exp(1/(13*x + 3*x*log(54*x^2 - 108*x - 12*x^3 + x^4 + 81) - 4*x^2))

________________________________________________________________________________________

sympy [A] time = 1.03, size = 36, normalized size = 1.44 \begin {gather*} x + e^{\frac {1}{- 4 x^{2} + 3 x \log {\left (x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81 \right )} + 13 x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+9)*ln(x**4-12*x**3+54*x**2-108*x+81)+8*x**2-49*x+39)*exp(1/(3*x*ln(x**4-12*x**3+54*x**2-108*

x+81)-4*x**2+13*x))+(9*x**3-27*x**2)*ln(x**4-12*x**3+54*x**2-108*x+81)**2+(-24*x**4+150*x**3-234*x**2)*ln(x**4

-12*x**3+54*x**2-108*x+81)+16*x**5-152*x**4+481*x**3-507*x**2)/((9*x**3-27*x**2)*ln(x**4-12*x**3+54*x**2-108*x

+81)**2+(-24*x**4+150*x**3-234*x**2)*ln(x**4-12*x**3+54*x**2-108*x+81)+16*x**5-152*x**4+481*x**3-507*x**2),x)

[Out]

x + exp(1/(-4*x**2 + 3*x*log(x**4 - 12*x**3 + 54*x**2 - 108*x + 81) + 13*x))

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]