\(\int \frac {1}{3} (6+2 x-432 x^3+2 x \log (x^2)) \, dx\) [7452]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=6+2 x-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right ) \]

[Out]

6-36*x^4+1/3*x^2*ln(x^2)+2*x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 2341} \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right )+2 x \]

[In]

Int[(6 + 2*x - 432*x^3 + 2*x*Log[x^2])/3,x]

[Out]

2*x - 36*x^4 + (x^2*Log[x^2])/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx \\ & = 2 x+\frac {x^2}{3}-36 x^4+\frac {2}{3} \int x \log \left (x^2\right ) \, dx \\ & = 2 x-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=2 x-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right ) \]

[In]

Integrate[(6 + 2*x - 432*x^3 + 2*x*Log[x^2])/3,x]

[Out]

2*x - 36*x^4 + (x^2*Log[x^2])/3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

method	result	size
default	\(2 x +\frac {x^{2} \ln \left (x^{2}\right )}{3}-36 x^{4}\)	\(19\)
norman	\(2 x +\frac {x^{2} \ln \left (x^{2}\right )}{3}-36 x^{4}\)	\(19\)
risch	\(2 x +\frac {x^{2} \ln \left (x^{2}\right )}{3}-36 x^{4}\)	\(19\)
parallelrisch	\(2 x +\frac {x^{2} \ln \left (x^{2}\right )}{3}-36 x^{4}\)	\(19\)
parts	\(2 x +\frac {x^{2} \ln \left (x^{2}\right )}{3}-36 x^{4}\)	\(19\)

[In]

int(2/3*x*ln(x^2)-144*x^3+2/3*x+2,x,method=_RETURNVERBOSE)

[Out]

2*x+1/3*x^2*ln(x^2)-36*x^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=-36 \, x^{4} + \frac {1}{3} \, x^{2} \log \left (x^{2}\right ) + 2 \, x \]

[In]

integrate(2/3*x*log(x^2)-144*x^3+2/3*x+2,x, algorithm="fricas")

[Out]

-36*x^4 + 1/3*x^2*log(x^2) + 2*x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=- 36 x^{4} + \frac {x^{2} \log {\left (x^{2} \right )}}{3} + 2 x \]

[In]

integrate(2/3*x*ln(x**2)-144*x**3+2/3*x+2,x)

[Out]

-36*x**4 + x**2*log(x**2)/3 + 2*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=-36 \, x^{4} + \frac {1}{3} \, x^{2} \log \left (x^{2}\right ) + 2 \, x \]

[In]

integrate(2/3*x*log(x^2)-144*x^3+2/3*x+2,x, algorithm="maxima")

[Out]

-36*x^4 + 1/3*x^2*log(x^2) + 2*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=-36 \, x^{4} + \frac {1}{3} \, x^{2} \log \left (x^{2}\right ) + 2 \, x \]

[In]

integrate(2/3*x*log(x^2)-144*x^3+2/3*x+2,x, algorithm="giac")

[Out]

-36*x^4 + 1/3*x^2*log(x^2) + 2*x

Mupad [B] (verification not implemented)

Time = 11.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{3} \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx=2\,x+\frac {x^2\,\ln \left (x^2\right )}{3}-36\,x^4 \]

[In]

int((2*x)/3 + (2*x*log(x^2))/3 - 144*x^3 + 2,x)

[Out]

2*x + (x^2*log(x^2))/3 - 36*x^4