3.97.78 \(\int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{2 x^2} \, dx\)

Optimal. Leaf size=24 \[ \frac {3+e^{72 \log ^2(3 x)}+e^x x}{2 x} \]

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Rubi [A]  time = 0.07, antiderivative size = 32, normalized size of antiderivative = 1.33, number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {12, 14, 2194, 2288} \begin {gather*} \frac {e^x}{2}+\frac {3}{2 x}+\frac {e^{72 \log ^2(3 x)}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + E^x*x^2 + E^(72*Log[3*x]^2)*(-1 + 144*Log[3*x]))/(2*x^2),x]

[Out]

E^x/2 + 3/(2*x) + E^(72*Log[3*x]^2)/(2*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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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} &=\frac {1}{2} \int \frac {-3+e^x x^2+e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {-3+e^x x^2}{x^2}+\frac {e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{x^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-3+e^x x^2}{x^2} \, dx+\frac {1}{2} \int \frac {e^{72 \log ^2(3 x)} (-1+144 \log (3 x))}{x^2} \, dx\\ &=\frac {e^{72 \log ^2(3 x)}}{2 x}+\frac {1}{2} \int \left (e^x-\frac {3}{x^2}\right ) \, dx\\ &=\frac {3}{2 x}+\frac {e^{72 \log ^2(3 x)}}{2 x}+\frac {\int e^x \, dx}{2}\\ &=\frac {e^x}{2}+\frac {3}{2 x}+\frac {e^{72 \log ^2(3 x)}}{2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 27, normalized size = 1.12 \begin {gather*} \frac {1}{2} \left (e^x+\frac {3}{x}+\frac {e^{72 \log ^2(3 x)}}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^x*x^2 + E^(72*Log[3*x]^2)*(-1 + 144*Log[3*x]))/(2*x^2),x]

[Out]

(E^x + 3/x + E^(72*Log[3*x]^2)/x)/2

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fricas [A]  time = 0.68, size = 20, normalized size = 0.83 \begin {gather*} \frac {x e^{x} + e^{\left (72 \, \log \left (3 \, x\right )^{2}\right )} + 3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((144*log(3*x)-1)*exp(72*log(3*x)^2)+exp(x)*x^2-3)/x^2,x, algorithm="fricas")

[Out]

1/2*(x*e^x + e^(72*log(3*x)^2) + 3)/x

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giac [A]  time = 0.17, size = 20, normalized size = 0.83 \begin {gather*} \frac {x e^{x} + e^{\left (72 \, \log \left (3 \, x\right )^{2}\right )} + 3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((144*log(3*x)-1)*exp(72*log(3*x)^2)+exp(x)*x^2-3)/x^2,x, algorithm="giac")

[Out]

1/2*(x*e^x + e^(72*log(3*x)^2) + 3)/x

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maple [A]  time = 0.04, size = 25, normalized size = 1.04

method	result	size
default	\(\frac {{\mathrm e}^{72 \ln \left (3 x \right )^{2}}}{2 x}+\frac {3}{2 x}+\frac {{\mathrm e}^{x}}{2}\)	\(25\)
risch	\(\frac {{\mathrm e}^{x} x +3}{2 x}+\frac {{\mathrm e}^{72 \ln \left (3 x \right )^{2}}}{2 x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((144*ln(3*x)-1)*exp(72*ln(3*x)^2)+exp(x)*x^2-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/2/x*exp(72*ln(3*x)^2)+3/2/x+1/2*exp(x)

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maxima [C]  time = 0.51, size = 116, normalized size = 4.83 \begin {gather*} \frac {1}{16} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (6 i \, \sqrt {2} \log \left (3 \, x\right ) - \frac {1}{24} i \, \sqrt {2}\right ) e^{\left (-\frac {1}{288}\right )} + \frac {1}{16} \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{12} \, \sqrt {\frac {1}{2}} \sqrt {-{\left (144 \, \log \left (3 \, x\right ) - 1\right )}^{2}}\right ) - 1\right )} {\left (144 \, \log \left (3 \, x\right ) - 1\right )}}{\sqrt {-{\left (144 \, \log \left (3 \, x\right ) - 1\right )}^{2}}} + 12 \, \sqrt {2} e^{\left (\frac {1}{288} \, {\left (144 \, \log \left (3 \, x\right ) - 1\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{288}\right )} + \frac {3}{2 \, x} + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((144*log(3*x)-1)*exp(72*log(3*x)^2)+exp(x)*x^2-3)/x^2,x, algorithm="maxima")

[Out]

1/16*I*sqrt(2)*sqrt(pi)*erf(6*I*sqrt(2)*log(3*x) - 1/24*I*sqrt(2))*e^(-1/288) + 1/16*sqrt(2)*(sqrt(2)*sqrt(1/2
)*sqrt(pi)*(erf(1/12*sqrt(1/2)*sqrt(-(144*log(3*x) - 1)^2)) - 1)*(144*log(3*x) - 1)/sqrt(-(144*log(3*x) - 1)^2
) + 12*sqrt(2)*e^(1/288*(144*log(3*x) - 1)^2))*e^(-1/288) + 3/2/x + 1/2*e^x

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mupad [B]  time = 6.39, size = 35, normalized size = 1.46 \begin {gather*} \frac {{\mathrm {e}}^x}{2}+\frac {3}{2\,x}+\frac {x^{144\,\ln \relax (3)}\,{\mathrm {e}}^{72\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{72\,{\ln \relax (3)}^2}}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2*exp(x))/2 + (exp(72*log(3*x)^2)*(144*log(3*x) - 1))/2 - 3/2)/x^2,x)

[Out]

exp(x)/2 + 3/(2*x) + (x^(144*log(3))*exp(72*log(x)^2)*exp(72*log(3)^2))/(2*x)

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sympy [A]  time = 0.45, size = 22, normalized size = 0.92 \begin {gather*} \frac {e^{x}}{2} + \frac {e^{72 \log {\left (3 x \right )}^{2}}}{2 x} + \frac {3}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((144*ln(3*x)-1)*exp(72*ln(3*x)**2)+exp(x)*x**2-3)/x**2,x)

[Out]

exp(x)/2 + exp(72*log(3*x)**2)/(2*x) + 3/(2*x)

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