3.63.92 \(\int \frac {1+(-3-x) \log ^2(3) \log ^2(3+x)}{(3+x) \log ^2(3) \log ^2(3+x)} \, dx\)

Optimal. Leaf size=23 \[ 5-x+\frac {-2-\frac {1}{\log (3+x)}+\log (\log (5))}{\log ^2(3)} \]

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Rubi [A]  time = 0.11, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {12, 6688, 2390, 2302, 30} \begin {gather*} -x-\frac {1}{\log ^2(3) \log (x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - 1/(Log[3]^2*Log[3 + 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 30

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

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.70 \begin {gather*} -x-\frac {1}{\log ^2(3) \log (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - 1/(Log[3]^2*Log[3 + x])

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fricas [A]  time = 0.96, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x \log \relax (3)^{2} \log \left (x + 3\right ) + 1}{\log \relax (3)^{2} \log \left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*log(3)^2*log(x + 3) + 1)/(log(3)^2*log(x + 3))

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giac [A]  time = 1.34, size = 19, normalized size = 0.83 \begin {gather*} -\frac {x \log \relax (3)^{2} + \frac {1}{\log \left (x + 3\right )}}{\log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*log(3)^2 + 1/log(x + 3))/log(3)^2

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maple [A]  time = 0.37, size = 17, normalized size = 0.74

method	result	size
risch	\(-x -\frac {1}{\ln \relax (3)^{2} \ln \left (3+x \right )}\)	\(17\)
derivativedivides	\(\frac {-\ln \relax (3)^{2} \left (3+x \right )-\frac {1}{\ln \left (3+x \right )}}{\ln \relax (3)^{2}}\)	\(24\)
default	\(\frac {-\ln \relax (3)^{2} \left (3+x \right )-\frac {1}{\ln \left (3+x \right )}}{\ln \relax (3)^{2}}\)	\(24\)
norman	\(\frac {-\frac {1}{\ln \relax (3)}-x \ln \relax (3) \ln \left (3+x \right )}{\ln \relax (3) \ln \left (3+x \right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x-1/ln(3)^2/ln(3+x)

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maxima [A]  time = 0.42, size = 36, normalized size = 1.57 \begin {gather*} -\frac {{\left (x - 3 \, \log \left (x + 3\right )\right )} \log \relax (3)^{2} + 3 \, \log \relax (3)^{2} \log \left (x + 3\right ) + \frac {1}{\log \left (x + 3\right )}}{\log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x - 3*log(x + 3))*log(3)^2 + 3*log(3)^2*log(x + 3) + 1/log(x + 3))/log(3)^2

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mupad [B]  time = 4.42, size = 16, normalized size = 0.70 \begin {gather*} -x-\frac {1}{\ln \left (x+3\right )\,{\ln \relax (3)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- x - 1/(log(x + 3)*log(3)^2)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.61 \begin {gather*} - x - \frac {1}{\log {\relax (3 )}^{2} \log {\left (x + 3 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 1/(log(3)**2*log(x + 3))

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