3.20.58 \(\int \frac {4 e^{5+\frac {2}{\log (x)}}+(-x+36 x^2) \log ^2(x)}{2 x \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ 2+e^5 \left (e^5-e^{\frac {2}{\log (x)}}\right )-\frac {x}{2}+9 x^2 \]

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Rubi [A]  time = 0.33, antiderivative size = 23, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {12, 6688, 2209} \begin {gather*} 9 x^2-\frac {x}{2}-e^{\frac {2}{\log (x)}+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^(5 + 2/Log[x]) + (-x + 36*x^2)*Log[x]^2)/(2*x*Log[x]^2),x]

[Out]

-E^(5 + 2/Log[x]) - x/2 + 9*x^2

Rule 12

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

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {4 e^{5+\frac {2}{\log (x)}}+\left (-x+36 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx\\ &=\frac {1}{2} \int \left (-1+36 x+\frac {4 e^{5+\frac {2}{\log (x)}}}{x \log ^2(x)}\right ) \, dx\\ &=-\frac {x}{2}+9 x^2+2 \int \frac {e^{5+\frac {2}{\log (x)}}}{x \log ^2(x)} \, dx\\ &=-\frac {x}{2}+9 x^2+2 \operatorname {Subst}\left (\int \frac {e^{5+\frac {2}{x}}}{x^2} \, dx,x,\log (x)\right )\\ &=-e^{5+\frac {2}{\log (x)}}-\frac {x}{2}+9 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 0.77 \begin {gather*} -e^{5+\frac {2}{\log (x)}}-\frac {x}{2}+9 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(5 + 2/Log[x]) + (-x + 36*x^2)*Log[x]^2)/(2*x*Log[x]^2),x]

[Out]

-E^(5 + 2/Log[x]) - x/2 + 9*x^2

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fricas [A]  time = 0.96, size = 23, normalized size = 0.77 \begin {gather*} 9 \, x^{2} - \frac {1}{2} \, x - e^{\left (\frac {5 \, \log \relax (x) + 2}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(5)*exp(2/log(x))+(36*x^2-x)*log(x)^2)/x/log(x)^2,x, algorithm="fricas")

[Out]

9*x^2 - 1/2*x - e^((5*log(x) + 2)/log(x))

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giac [A]  time = 0.32, size = 20, normalized size = 0.67 \begin {gather*} 9 \, x^{2} - \frac {1}{2} \, x - e^{\left (\frac {2}{\log \relax (x)} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(5)*exp(2/log(x))+(36*x^2-x)*log(x)^2)/x/log(x)^2,x, algorithm="giac")

[Out]

9*x^2 - 1/2*x - e^(2/log(x) + 5)

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maple [A]  time = 0.06, size = 21, normalized size = 0.70

method	result	size
default	\(9 x^{2}-\frac {x}{2}-{\mathrm e}^{5} {\mathrm e}^{\frac {2}{\ln \relax (x )}}\)	\(21\)
risch	\(9 x^{2}-\frac {x}{2}-{\mathrm e}^{\frac {5 \ln \relax (x )+2}{\ln \relax (x )}}\)	\(24\)
norman	\(\frac {-\frac {x \ln \relax (x )}{2}+9 x^{2} \ln \relax (x )-{\mathrm e}^{5} \ln \relax (x ) {\mathrm e}^{\frac {2}{\ln \relax (x )}}}{\ln \relax (x )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(4*exp(5)*exp(2/ln(x))+(36*x^2-x)*ln(x)^2)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

9*x^2-1/2*x-exp(5)*exp(2/ln(x))

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maxima [A]  time = 0.48, size = 20, normalized size = 0.67 \begin {gather*} 9 \, x^{2} - \frac {1}{2} \, x - e^{\left (\frac {2}{\log \relax (x)} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(5)*exp(2/log(x))+(36*x^2-x)*log(x)^2)/x/log(x)^2,x, algorithm="maxima")

[Out]

9*x^2 - 1/2*x - e^(2/log(x) + 5)

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mupad [B]  time = 1.24, size = 20, normalized size = 0.67 \begin {gather*} 9\,x^2-{\mathrm {e}}^{\frac {2}{\ln \relax (x)}}\,{\mathrm {e}}^5-\frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((log(x)^2*(x - 36*x^2))/2 - 2*exp(2/log(x))*exp(5))/(x*log(x)^2),x)

[Out]

9*x^2 - exp(2/log(x))*exp(5) - x/2

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sympy [A]  time = 0.28, size = 17, normalized size = 0.57 \begin {gather*} 9 x^{2} - \frac {x}{2} - e^{5} e^{\frac {2}{\log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(5)*exp(2/ln(x))+(36*x**2-x)*ln(x)**2)/x/ln(x)**2,x)

[Out]

9*x**2 - x/2 - exp(5)*exp(2/log(x))

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