\(\int \frac {-9-24 x^2+4 x^3+e^{x^2} (-3+6 x^2)-24 x^2 \log (x)}{24 x^2} \, dx\) [6863]

Optimal result

Integrand size = 39, antiderivative size = 32 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {1}{4} \left (\frac {3+e^{x^2}}{2 x}+\frac {x^2}{3}\right )-x \log (x) \]

[Out]

1/12*x^2+1/8/x*(exp(x^2)+3)-x*ln(x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 14, 2326, 2332} \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {x^2}{12}+\frac {e^{x^2}}{8 x}+\frac {3}{8 x}-x \log (x) \]

[In]

Int[(-9 - 24*x^2 + 4*x^3 + E^x^2*(-3 + 6*x^2) - 24*x^2*Log[x])/(24*x^2),x]

[Out]

3/(8*x) + E^x^2/(8*x) + x^2/12 - x*Log[x]

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{24} \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{x^2} \, dx \\ & = \frac {1}{24} \int \left (\frac {3 e^{x^2} \left (-1+2 x^2\right )}{x^2}+\frac {-9-24 x^2+4 x^3-24 x^2 \log (x)}{x^2}\right ) \, dx \\ & = \frac {1}{24} \int \frac {-9-24 x^2+4 x^3-24 x^2 \log (x)}{x^2} \, dx+\frac {1}{8} \int \frac {e^{x^2} \left (-1+2 x^2\right )}{x^2} \, dx \\ & = \frac {e^{x^2}}{8 x}+\frac {1}{24} \int \left (\frac {-9-24 x^2+4 x^3}{x^2}-24 \log (x)\right ) \, dx \\ & = \frac {e^{x^2}}{8 x}+\frac {1}{24} \int \frac {-9-24 x^2+4 x^3}{x^2} \, dx-\int \log (x) \, dx \\ & = \frac {e^{x^2}}{8 x}+x-x \log (x)+\frac {1}{24} \int \left (-24-\frac {9}{x^2}+4 x\right ) \, dx \\ & = \frac {3}{8 x}+\frac {e^{x^2}}{8 x}+\frac {x^2}{12}-x \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {1}{24} \left (\frac {9}{x}+\frac {3 e^{x^2}}{x}+2 x^2-24 x \log (x)\right ) \]

[In]

Integrate[(-9 - 24*x^2 + 4*x^3 + E^x^2*(-3 + 6*x^2) - 24*x^2*Log[x])/(24*x^2),x]

[Out]

(9/x + (3*E^x^2)/x + 2*x^2 - 24*x*Log[x])/24

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
norman	\(\frac {\frac {3}{8}+\frac {x^{3}}{12}-x^{2} \ln \left (x \right )+\frac {{\mathrm e}^{x^{2}}}{8}}{x}\)	\(25\)
risch	\(-x \ln \left (x \right )+\frac {2 x^{3}+3 \,{\mathrm e}^{x^{2}}+9}{24 x}\)	\(25\)
default	\(\frac {x^{2}}{12}+\frac {3}{8 x}+\frac {{\mathrm e}^{x^{2}}}{8 x}-x \ln \left (x \right )\)	\(26\)
parallelrisch	\(\frac {-24 x^{2} \ln \left (x \right )+2 x^{3}+3 \,{\mathrm e}^{x^{2}}+9}{24 x}\)	\(26\)
parts	\(\frac {x^{2}}{12}+\frac {3}{8 x}+\frac {{\mathrm e}^{x^{2}}}{8 x}-x \ln \left (x \right )\)	\(26\)

[In]

int(1/24*(-24*x^2*ln(x)+(6*x^2-3)*exp(x^2)+4*x^3-24*x^2-9)/x^2,x,method=_RETURNVERBOSE)

[Out]

(3/8+1/12*x^3-x^2*ln(x)+1/8*exp(x^2))/x

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {2 \, x^{3} - 24 \, x^{2} \log \left (x\right ) + 3 \, e^{\left (x^{2}\right )} + 9}{24 \, x} \]

[In]

integrate(1/24*(-24*x^2*log(x)+(6*x^2-3)*exp(x^2)+4*x^3-24*x^2-9)/x^2,x, algorithm="fricas")

[Out]

1/24*(2*x^3 - 24*x^2*log(x) + 3*e^(x^2) + 9)/x

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {x^{2}}{12} - x \log {\left (x \right )} + \frac {e^{x^{2}}}{8 x} + \frac {3}{8 x} \]

[In]

integrate(1/24*(-24*x**2*ln(x)+(6*x**2-3)*exp(x**2)+4*x**3-24*x**2-9)/x**2,x)

[Out]

x**2/12 - x*log(x) + exp(x**2)/(8*x) + 3/(8*x)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {1}{12} \, x^{2} - x \log \left (x\right ) - \frac {1}{8} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) + \frac {\sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{16 \, x} + \frac {3}{8 \, x} \]

[In]

integrate(1/24*(-24*x^2*log(x)+(6*x^2-3)*exp(x^2)+4*x^3-24*x^2-9)/x^2,x, algorithm="maxima")

[Out]

1/12*x^2 - x*log(x) - 1/8*I*sqrt(pi)*erf(I*x) + 1/16*sqrt(-x^2)*gamma(-1/2, -x^2)/x + 3/8/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {2 \, x^{3} - 24 \, x^{2} \log \left (x\right ) + 3 \, e^{\left (x^{2}\right )} + 9}{24 \, x} \]

[In]

integrate(1/24*(-24*x^2*log(x)+(6*x^2-3)*exp(x^2)+4*x^3-24*x^2-9)/x^2,x, algorithm="giac")

[Out]

1/24*(2*x^3 - 24*x^2*log(x) + 3*e^(x^2) + 9)/x

Mupad [B] (verification not implemented)

Time = 12.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {-9-24 x^2+4 x^3+e^{x^2} \left (-3+6 x^2\right )-24 x^2 \log (x)}{24 x^2} \, dx=\frac {\frac {{\mathrm {e}}^{x^2}}{8}+\frac {3}{8}}{x}-x\,\ln \left (x\right )+\frac {x^2}{12} \]

[In]

int(-(x^2*log(x) - (exp(x^2)*(6*x^2 - 3))/24 + x^2 - x^3/6 + 3/8)/x^2,x)

[Out]

(exp(x^2)/8 + 3/8)/x - x*log(x) + x^2/12