3.9.88 \(\int \frac {-3+(6+x+\log (2)+e^{2 x} (12+2 x+2 \log (2))+(-12-2 x-2 \log (2)) \log (25)) \log ^2(6+x+\log (2))}{(12+2 x+2 \log (2)) \log ^2(6+x+\log (2))} \, dx\)

Optimal. Leaf size=27 \[ -x \log (25)+\frac {1}{2} \left (e^{2 x}+x+\frac {3}{\log (6+x+\log (2))}\right ) \]

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Rubi [A]  time = 0.37, antiderivative size = 33, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6688, 2194, 2390, 2302, 30} \begin {gather*} \frac {e^{2 x}}{2}+\frac {1}{2} x (1-2 \log (25))+\frac {3}{2 \log (x+6+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + (6 + x + Log[2] + E^(2*x)*(12 + 2*x + 2*Log[2]) + (-12 - 2*x - 2*Log[2])*Log[25])*Log[6 + x + Log[2]
]^2)/((12 + 2*x + 2*Log[2])*Log[6 + x + Log[2]]^2),x]

[Out]

E^(2*x)/2 + (x*(1 - 2*Log[25]))/2 + 3/(2*Log[6 + x + Log[2]])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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} &=\int \left (\frac {1}{2}+e^{2 x}-\log (25)-\frac {3}{2 (6+x+\log (2)) \log ^2(6+x+\log (2))}\right ) \, dx\\ &=\frac {1}{2} x (1-2 \log (25))-\frac {3}{2} \int \frac {1}{(6+x+\log (2)) \log ^2(6+x+\log (2))} \, dx+\int e^{2 x} \, dx\\ &=\frac {e^{2 x}}{2}+\frac {1}{2} x (1-2 \log (25))-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,6+x+\log (2)\right )\\ &=\frac {e^{2 x}}{2}+\frac {1}{2} x (1-2 \log (25))-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (6+x+\log (2))\right )\\ &=\frac {e^{2 x}}{2}+\frac {1}{2} x (1-2 \log (25))+\frac {3}{2 \log (6+x+\log (2))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 32, normalized size = 1.19 \begin {gather*} \frac {e^{2 x}}{2}+\frac {x}{2}-x \log (25)+\frac {3}{2 \log (6+x+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + (6 + x + Log[2] + E^(2*x)*(12 + 2*x + 2*Log[2]) + (-12 - 2*x - 2*Log[2])*Log[25])*Log[6 + x +
Log[2]]^2)/((12 + 2*x + 2*Log[2])*Log[6 + x + Log[2]]^2),x]

[Out]

E^(2*x)/2 + x/2 - x*Log[25] + 3/(2*Log[6 + x + Log[2]])

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fricas [A]  time = 1.10, size = 34, normalized size = 1.26 \begin {gather*} -\frac {{\left (4 \, x \log \relax (5) - x - e^{\left (2 \, x\right )}\right )} \log \left (x + \log \relax (2) + 6\right ) - 3}{2 \, \log \left (x + \log \relax (2) + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*log(2)+2*x+12)*exp(x)^2+2*(-2*log(2)-2*x-12)*log(5)+log(2)+x+6)*log(log(2)+x+6)^2-3)/(2*log(2)+
2*x+12)/log(log(2)+x+6)^2,x, algorithm="fricas")

[Out]

-1/2*((4*x*log(5) - x - e^(2*x))*log(x + log(2) + 6) - 3)/log(x + log(2) + 6)

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giac [A]  time = 0.67, size = 44, normalized size = 1.63 \begin {gather*} -\frac {4 \, x \log \relax (5) \log \left (x + \log \relax (2) + 6\right ) - x \log \left (x + \log \relax (2) + 6\right ) - e^{\left (2 \, x\right )} \log \left (x + \log \relax (2) + 6\right ) - 3}{2 \, \log \left (x + \log \relax (2) + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*log(2)+2*x+12)*exp(x)^2+2*(-2*log(2)-2*x-12)*log(5)+log(2)+x+6)*log(log(2)+x+6)^2-3)/(2*log(2)+
2*x+12)/log(log(2)+x+6)^2,x, algorithm="giac")

[Out]

-1/2*(4*x*log(5)*log(x + log(2) + 6) - x*log(x + log(2) + 6) - e^(2*x)*log(x + log(2) + 6) - 3)/log(x + log(2)
+ 6)

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maple [A]  time = 0.20, size = 26, normalized size = 0.96

method	result	size
default	\(\frac {3}{2 \ln \left (\ln \relax (2)+x +6\right )}+\frac {{\mathrm e}^{2 x}}{2}+\frac {x}{2}-2 x \ln \relax (5)\)	\(26\)
risch	\(\frac {3}{2 \ln \left (\ln \relax (2)+x +6\right )}+\frac {{\mathrm e}^{2 x}}{2}+\frac {x}{2}-2 x \ln \relax (5)\)	\(26\)
norman	\(\frac {\frac {3}{2}+\left (-2 \ln \relax (5)+\frac {1}{2}\right ) x \ln \left (\ln \relax (2)+x +6\right )+\frac {{\mathrm e}^{2 x} \ln \left (\ln \relax (2)+x +6\right )}{2}}{\ln \left (\ln \relax (2)+x +6\right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*ln(2)+2*x+12)*exp(x)^2+2*(-2*ln(2)-2*x-12)*ln(5)+ln(2)+x+6)*ln(ln(2)+x+6)^2-3)/(2*ln(2)+2*x+12)/ln(ln
(2)+x+6)^2,x,method=_RETURNVERBOSE)

[Out]

3/2/ln(ln(2)+x+6)+1/2*exp(x)^2+1/2*x-2*x*ln(5)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -e^{\left (-2 \, \log \relax (2) - 12\right )} E_{1}\left (-2 \, x - 2 \, \log \relax (2) - 12\right ) \log \relax (2) - 2 \, \log \relax (5) \log \relax (2) \log \left (x + \log \relax (2) + 6\right ) - 6 \, e^{\left (-2 \, \log \relax (2) - 12\right )} E_{1}\left (-2 \, x - 2 \, \log \relax (2) - 12\right ) - {\left (\log \relax (2) + 6\right )} \int \frac {e^{\left (2 \, x\right )}}{2 \, {\left (x^{2} + 2 \, x {\left (\log \relax (2) + 6\right )} + \log \relax (2)^{2} + 12 \, \log \relax (2) + 36\right )}}\,{d x} + 2 \, {\left ({\left (\log \relax (2) + 6\right )} \log \left (x + \log \relax (2) + 6\right ) - x\right )} \log \relax (5) - \frac {1}{2} \, {\left (\log \relax (2) + 6\right )} \log \left (x + \log \relax (2) + 6\right ) - 12 \, \log \relax (5) \log \left (x + \log \relax (2) + 6\right ) + \frac {1}{2} \, \log \relax (2) \log \left (x + \log \relax (2) + 6\right ) + \frac {1}{2} \, x + \frac {x e^{\left (2 \, x\right )}}{2 \, {\left (x + \log \relax (2) + 6\right )}} + \frac {3}{2 \, \log \left (x + \log \relax (2) + 6\right )} + 3 \, \log \left (x + \log \relax (2) + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*log(2)+2*x+12)*exp(x)^2+2*(-2*log(2)-2*x-12)*log(5)+log(2)+x+6)*log(log(2)+x+6)^2-3)/(2*log(2)+
2*x+12)/log(log(2)+x+6)^2,x, algorithm="maxima")

[Out]

-e^(-2*log(2) - 12)*exp_integral_e(1, -2*x - 2*log(2) - 12)*log(2) - 2*log(5)*log(2)*log(x + log(2) + 6) - 6*e
^(-2*log(2) - 12)*exp_integral_e(1, -2*x - 2*log(2) - 12) - (log(2) + 6)*integrate(1/2*e^(2*x)/(x^2 + 2*x*(log
(2) + 6) + log(2)^2 + 12*log(2) + 36), x) + 2*((log(2) + 6)*log(x + log(2) + 6) - x)*log(5) - 1/2*(log(2) + 6)
*log(x + log(2) + 6) - 12*log(5)*log(x + log(2) + 6) + 1/2*log(2)*log(x + log(2) + 6) + 1/2*x + 1/2*x*e^(2*x)/
(x + log(2) + 6) + 3/2/log(x + log(2) + 6) + 3*log(x + log(2) + 6)

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mupad [B]  time = 0.51, size = 38, normalized size = 1.41 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{2}-x\,\left (\ln \left (25\right )-\frac {1}{2}\right )+\frac {3\,\left (x+\ln \relax (2)+6\right )}{\ln \left (x+\ln \relax (2)+6\right )\,\left (2\,x+\ln \relax (4)+12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + log(2) + 6)^2*(x + log(2) + exp(2*x)*(2*x + 2*log(2) + 12) - 2*log(5)*(2*x + 2*log(2) + 12) + 6)
- 3)/(log(x + log(2) + 6)^2*(2*x + 2*log(2) + 12)),x)

[Out]

exp(2*x)/2 - x*(log(25) - 1/2) + (3*(x + log(2) + 6))/(log(x + log(2) + 6)*(2*x + log(4) + 12))

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sympy [A]  time = 0.37, size = 27, normalized size = 1.00 \begin {gather*} x \left (\frac {1}{2} - 2 \log {\relax (5 )}\right ) + \frac {e^{2 x}}{2} + \frac {3}{2 \log {\left (x + \log {\relax (2 )} + 6 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*ln(2)+2*x+12)*exp(x)**2+2*(-2*ln(2)-2*x-12)*ln(5)+ln(2)+x+6)*ln(ln(2)+x+6)**2-3)/(2*ln(2)+2*x+1
2)/ln(ln(2)+x+6)**2,x)

[Out]

x*(1/2 - 2*log(5)) + exp(2*x)/2 + 3/(2*log(x + log(2) + 6))

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