3.58.44 \(\int \frac {-36 e^{\frac {6}{x^2}}+54 x^4+e^{\frac {3}{x^2}} (108 x-18 x^3)}{x^3} \, dx\)

Optimal. Leaf size=15 \[ 3 \left (e^{\frac {3}{x^2}}-3 x\right )^2 \]

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Rubi [A]  time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.67, number of steps used = 4, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14, 2209, 2288} \begin {gather*} 27 x^2-18 e^{\frac {3}{x^2}} x+3 e^{\frac {6}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36*E^(6/x^2) + 54*x^4 + E^(3/x^2)*(108*x - 18*x^3))/x^3,x]

[Out]

3*E^(6/x^2) - 18*E^(3/x^2)*x + 27*x^2

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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2288

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {36 e^{\frac {6}{x^2}}}{x^3}+54 x-\frac {18 e^{\frac {3}{x^2}} \left (-6+x^2\right )}{x^2}\right ) \, dx\\ &=27 x^2-18 \int \frac {e^{\frac {3}{x^2}} \left (-6+x^2\right )}{x^2} \, dx-36 \int \frac {e^{\frac {6}{x^2}}}{x^3} \, dx\\ &=3 e^{\frac {6}{x^2}}-18 e^{\frac {3}{x^2}} x+27 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} 3 \left (e^{\frac {3}{x^2}}-3 x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*E^(6/x^2) + 54*x^4 + E^(3/x^2)*(108*x - 18*x^3))/x^3,x]

[Out]

3*(E^(3/x^2) - 3*x)^2

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fricas [A]  time = 0.55, size = 23, normalized size = 1.53 \begin {gather*} 27 \, x^{2} - 18 \, x e^{\left (\frac {3}{x^{2}}\right )} + 3 \, e^{\left (\frac {6}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*exp(3/x^2)^2+(-18*x^3+108*x)*exp(3/x^2)+54*x^4)/x^3,x, algorithm="fricas")

[Out]

27*x^2 - 18*x*e^(3/x^2) + 3*e^(6/x^2)

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giac [A]  time = 0.24, size = 23, normalized size = 1.53 \begin {gather*} 27 \, x^{2} - 18 \, x e^{\left (\frac {3}{x^{2}}\right )} + 3 \, e^{\left (\frac {6}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*exp(3/x^2)^2+(-18*x^3+108*x)*exp(3/x^2)+54*x^4)/x^3,x, algorithm="giac")

[Out]

27*x^2 - 18*x*e^(3/x^2) + 3*e^(6/x^2)

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maple [A]  time = 0.09, size = 24, normalized size = 1.60

method	result	size
derivativedivides	\(27 x^{2}-18 x \,{\mathrm e}^{\frac {3}{x^{2}}}+3 \,{\mathrm e}^{\frac {6}{x^{2}}}\)	\(24\)
default	\(27 x^{2}-18 x \,{\mathrm e}^{\frac {3}{x^{2}}}+3 \,{\mathrm e}^{\frac {6}{x^{2}}}\)	\(24\)
risch	\(27 x^{2}-18 x \,{\mathrm e}^{\frac {3}{x^{2}}}+3 \,{\mathrm e}^{\frac {6}{x^{2}}}\)	\(24\)
norman	\(\frac {27 x^{4}-18 \,{\mathrm e}^{\frac {3}{x^{2}}} x^{3}+3 \,{\mathrm e}^{\frac {6}{x^{2}}} x^{2}}{x^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-36*exp(3/x^2)^2+(-18*x^3+108*x)*exp(3/x^2)+54*x^4)/x^3,x,method=_RETURNVERBOSE)

[Out]

27*x^2+3*exp(1/x^2)^6-18*exp(1/x^2)^3*x

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maxima [C]  time = 0.37, size = 66, normalized size = 4.40 \begin {gather*} -9 \, \sqrt {3} x \sqrt {-\frac {1}{x^{2}}} \Gamma \left (-\frac {1}{2}, -\frac {3}{x^{2}}\right ) + 27 \, x^{2} - \frac {18 \, \sqrt {3} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {3} \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )}}{x \sqrt {-\frac {1}{x^{2}}}} + 3 \, e^{\left (\frac {6}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*exp(3/x^2)^2+(-18*x^3+108*x)*exp(3/x^2)+54*x^4)/x^3,x, algorithm="maxima")

[Out]

-9*sqrt(3)*x*sqrt(-1/x^2)*gamma(-1/2, -3/x^2) + 27*x^2 - 18*sqrt(3)*sqrt(pi)*(erf(sqrt(3)*sqrt(-1/x^2)) - 1)/(
x*sqrt(-1/x^2)) + 3*e^(6/x^2)

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mupad [B]  time = 3.62, size = 16, normalized size = 1.07 \begin {gather*} 3\,{\left (3\,x-{\mathrm {e}}^{\frac {3}{x^2}}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3/x^2)*(108*x - 18*x^3) - 36*exp(6/x^2) + 54*x^4)/x^3,x)

[Out]

3*(3*x - exp(3/x^2))^2

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sympy [A]  time = 0.13, size = 22, normalized size = 1.47 \begin {gather*} 27 x^{2} - 18 x e^{\frac {3}{x^{2}}} + 3 e^{\frac {6}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*exp(3/x**2)**2+(-18*x**3+108*x)*exp(3/x**2)+54*x**4)/x**3,x)

[Out]

27*x**2 - 18*x*exp(3/x**2) + 3*exp(6/x**2)

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