3.86.74 \(\int \frac {3+2 x-36 x^3-12 x^4+e^4 (-1+12 x^3)+(-36 x^3+12 e^4 x^3-12 x^4) \log (3 x-e^4 x+x^2)+(-9 x^3+3 e^4 x^3-3 x^4) \log ^2(3 x-e^4 x+x^2)}{-12 x+4 e^4 x-4 x^2+(-12 x+4 e^4 x-4 x^2) \log (3 x-e^4 x+x^2)+(-3 x+e^4 x-x^2) \log ^2(3 x-e^4 x+x^2)} \, dx\)

Optimal. Leaf size=19 \[ x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )} \]

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Rubi [A]  time = 0.85, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 175, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6, 6688, 6742, 6686} \begin {gather*} x^3+\frac {1}{\log \left (x \left (x-e^4+3\right )\right )+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x - 36*x^3 - 12*x^4 + E^4*(-1 + 12*x^3) + (-36*x^3 + 12*E^4*x^3 - 12*x^4)*Log[3*x - E^4*x + x^2] +
(-9*x^3 + 3*E^4*x^3 - 3*x^4)*Log[3*x - E^4*x + x^2]^2)/(-12*x + 4*E^4*x - 4*x^2 + (-12*x + 4*E^4*x - 4*x^2)*Lo
g[3*x - E^4*x + x^2] + (-3*x + E^4*x - x^2)*Log[3*x - E^4*x + x^2]^2),x]

[Out]

x^3 + (2 + Log[x*(3 - E^4 + x)])^(-1)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{\left (-12+4 e^4\right ) x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx\\ &=\int \frac {-3-2 x+36 x^3+12 x^4+e^4 \left (1-12 x^3\right )+12 x^3 \left (3-e^4+x\right ) \log \left (x \left (3-e^4+x\right )\right )+3 x^3 \left (3-e^4+x\right ) \log ^2\left (x \left (3-e^4+x\right )\right )}{x \left (3-e^4+x\right ) \left (2+\log \left (x \left (3-e^4+x\right )\right )\right )^2} \, dx\\ &=\int \left (3 x^2+\frac {3-e^4+2 x}{\left (-3+e^4-x\right ) x \left (2+\log \left (x \left (3-e^4+x\right )\right )\right )^2}\right ) \, dx\\ &=x^3+\int \frac {3-e^4+2 x}{\left (-3+e^4-x\right ) x \left (2+\log \left (x \left (3-e^4+x\right )\right )\right )^2} \, dx\\ &=x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x - 36*x^3 - 12*x^4 + E^4*(-1 + 12*x^3) + (-36*x^3 + 12*E^4*x^3 - 12*x^4)*Log[3*x - E^4*x + x
^2] + (-9*x^3 + 3*E^4*x^3 - 3*x^4)*Log[3*x - E^4*x + x^2]^2)/(-12*x + 4*E^4*x - 4*x^2 + (-12*x + 4*E^4*x - 4*x
^2)*Log[3*x - E^4*x + x^2] + (-3*x + E^4*x - x^2)*Log[3*x - E^4*x + x^2]^2),x]

[Out]

x^3 + (2 + Log[x*(3 - E^4 + x)])^(-1)

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fricas [B]  time = 0.62, size = 42, normalized size = 2.21 \begin {gather*} \frac {x^{3} \log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2 \, x^{3} + 1}{\log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp(4)-12*x^4-36*x^3)*log(-x*exp(4)+x^2
+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+2*x+3)/((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12
*x)*log(-x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.32, size = 42, normalized size = 2.21 \begin {gather*} \frac {x^{3} \log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2 \, x^{3} + 1}{\log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp(4)-12*x^4-36*x^3)*log(-x*exp(4)+x^2
+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+2*x+3)/((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12
*x)*log(-x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.14, size = 22, normalized size = 1.16

method	result	size
risch	\(x^{3}+\frac {1}{\ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2}\)	\(22\)
norman	\(\frac {x^{3} \ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2 x^{3}+1}{\ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3*exp(4)-3*x^4-9*x^3)*ln(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp(4)-12*x^4-36*x^3)*ln(-x*exp(4)+x^2+3*x)+(1
2*x^3-1)*exp(4)-12*x^4-36*x^3+2*x+3)/((x*exp(4)-x^2-3*x)*ln(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12*x)*ln(-x
*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.40, size = 40, normalized size = 2.11 \begin {gather*} \frac {x^{3} \log \left (x - e^{4} + 3\right ) + x^{3} \log \relax (x) + 2 \, x^{3} + 1}{\log \left (x - e^{4} + 3\right ) + \log \relax (x) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp(4)-12*x^4-36*x^3)*log(-x*exp(4)+x^2
+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+2*x+3)/((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12
*x)*log(-x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.06, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{\ln \left (3\,x-x\,{\mathrm {e}}^4+x^2\right )+2}+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3*x - x*exp(4) + x^2)*(36*x^3 - 12*x^3*exp(4) + 12*x^4) - exp(4)*(12*x^3 - 1) - 2*x + 36*x^3 + 12*x^4
+ log(3*x - x*exp(4) + x^2)^2*(9*x^3 - 3*x^3*exp(4) + 3*x^4) - 3)/(12*x - 4*x*exp(4) + log(3*x - x*exp(4) + x
^2)*(12*x - 4*x*exp(4) + 4*x^2) + log(3*x - x*exp(4) + x^2)^2*(3*x - x*exp(4) + x^2) + 4*x^2),x)

[Out]

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

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sympy [A]  time = 0.20, size = 19, normalized size = 1.00 \begin {gather*} x^{3} + \frac {1}{\log {\left (x^{2} - x e^{4} + 3 x \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3*exp(4)-3*x**4-9*x**3)*ln(-x*exp(4)+x**2+3*x)**2+(12*x**3*exp(4)-12*x**4-36*x**3)*ln(-x*exp(
4)+x**2+3*x)+(12*x**3-1)*exp(4)-12*x**4-36*x**3+2*x+3)/((x*exp(4)-x**2-3*x)*ln(-x*exp(4)+x**2+3*x)**2+(4*x*exp
(4)-4*x**2-12*x)*ln(-x*exp(4)+x**2+3*x)+4*x*exp(4)-4*x**2-12*x),x)

[Out]

x**3 + 1/(log(x**2 - x*exp(4) + 3*x) + 2)

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