3.102.41 \(\int \frac {6+x^2+(6+x^2) \log (24+4 x^2)+\frac {e^{e^2+x} (-6+2 x-x^2+(-6-x^2) \log (24+4 x^2))}{1+\log (24+4 x^2)}}{6+x^2+(6+x^2) \log (24+4 x^2)} \, dx\)

Optimal. Leaf size=27 \[ x-\log (2)-\frac {e^{e^2+x}}{1+\log \left (4 \left (6+x^2\right )\right )} \]

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Rubi [B]  time = 1.32, antiderivative size = 57, normalized size of antiderivative = 2.11, number of steps used = 4, number of rules used = 3, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6741, 6725, 2288} \begin {gather*} x-\frac {e^{x+e^2} \left (x^2+x^2 \log \left (4 \left (x^2+6\right )\right )+6 \log \left (4 \left (x^2+6\right )\right )+6\right )}{\left (x^2+6\right ) \left (\log \left (4 \left (x^2+6\right )\right )+1\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + x^2 + (6 + x^2)*Log[24 + 4*x^2] + (E^(E^2 + x)*(-6 + 2*x - x^2 + (-6 - x^2)*Log[24 + 4*x^2]))/(1 + Lo
g[24 + 4*x^2]))/(6 + x^2 + (6 + x^2)*Log[24 + 4*x^2]),x]

[Out]

x - (E^(E^2 + x)*(6 + x^2 + 6*Log[4*(6 + x^2)] + x^2*Log[4*(6 + x^2)]))/((6 + x^2)*(1 + Log[4*(6 + x^2)])^2)

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]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+x^2+\left (6+x^2\right ) \log \left (24+4 x^2\right )+\frac {e^{e^2+x} \left (-6+2 x-x^2+\left (-6-x^2\right ) \log \left (24+4 x^2\right )\right )}{1+\log \left (24+4 x^2\right )}}{\left (6+x^2\right ) \left (1+\log \left (4 \left (6+x^2\right )\right )\right )} \, dx\\ &=\int \left (1-\frac {e^{e^2+x} \left (6-2 x+x^2+6 \log \left (4 \left (6+x^2\right )\right )+x^2 \log \left (4 \left (6+x^2\right )\right )\right )}{\left (6+x^2\right ) \left (1+\log \left (4 \left (6+x^2\right )\right )\right )^2}\right ) \, dx\\ &=x-\int \frac {e^{e^2+x} \left (6-2 x+x^2+6 \log \left (4 \left (6+x^2\right )\right )+x^2 \log \left (4 \left (6+x^2\right )\right )\right )}{\left (6+x^2\right ) \left (1+\log \left (4 \left (6+x^2\right )\right )\right )^2} \, dx\\ &=x-\frac {e^{e^2+x} \left (6+x^2+6 \log \left (4 \left (6+x^2\right )\right )+x^2 \log \left (4 \left (6+x^2\right )\right )\right )}{\left (6+x^2\right ) \left (1+\log \left (4 \left (6+x^2\right )\right )\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 23, normalized size = 0.85 \begin {gather*} x-\frac {e^{e^2+x}}{1+\log \left (4 \left (6+x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - E^(E^2 + x)/(1 + Log[4*(6 + x^2)])

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fricas [A]  time = 0.68, size = 22, normalized size = 0.81 \begin {gather*} x - e^{\left (x + e^{2} - \log \left (\log \left (4 \, x^{2} + 24\right ) + 1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - e^(x + e^2 - log(log(4*x^2 + 24) + 1))

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giac [A]  time = 0.27, size = 37, normalized size = 1.37 \begin {gather*} \frac {2 \, x \log \relax (2) + x \log \left (x^{2} + 6\right ) + x - e^{\left (x + e^{2}\right )}}{2 \, \log \relax (2) + \log \left (x^{2} + 6\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*log(2) + x*log(x^2 + 6) + x - e^(x + e^2))/(2*log(2) + log(x^2 + 6) + 1)

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maple [A]  time = 0.31, size = 22, normalized size = 0.81

method	result	size
risch	\(x -\frac {{\mathrm e}^{x +{\mathrm e}^{2}}}{\ln \left (4 x^{2}+24\right )+1}\)	\(22\)
default	\(x -\frac {{\mathrm e}^{x +{\mathrm e}^{2}}}{2 \ln \relax (2)+\ln \left (x^{2}+6\right )+1}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-1/(ln(4*x^2+24)+1)*exp(x+exp(2))

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maxima [A]  time = 0.47, size = 39, normalized size = 1.44 \begin {gather*} \frac {x {\left (2 \, \log \relax (2) + 1\right )} + x \log \left (x^{2} + 6\right ) - e^{\left (x + e^{2}\right )}}{2 \, \log \relax (2) + \log \left (x^{2} + 6\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(2*log(2) + 1) + x*log(x^2 + 6) - e^(x + e^2))/(2*log(2) + log(x^2 + 6) + 1)

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mupad [B]  time = 7.60, size = 21, normalized size = 0.78 \begin {gather*} x-\frac {{\mathrm {e}}^{{\mathrm {e}}^2}\,{\mathrm {e}}^x}{\ln \left (4\,x^2+24\right )+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4*x^2 + 24)*(x^2 + 6) - exp(x + exp(2) - log(log(4*x^2 + 24) + 1))*(log(4*x^2 + 24)*(x^2 + 6) - 2*x +
x^2 + 6) + x^2 + 6)/(log(4*x^2 + 24)*(x^2 + 6) + x^2 + 6),x)

[Out]

x - (exp(exp(2))*exp(x))/(log(4*x^2 + 24) + 1)

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sympy [A]  time = 0.35, size = 17, normalized size = 0.63 \begin {gather*} x - \frac {e^{x + e^{2}}}{\log {\left (4 x^{2} + 24 \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - exp(x + exp(2))/(log(4*x**2 + 24) + 1)

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