\(\int \frac {13+(68 x+26 x^2) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx\) [2826]

Optimal result

Integrand size = 34, antiderivative size = 14 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^2+\log (\log (-2+x+12 (3+x))) \]

[Out]

x^2+ln(ln(13*x+34))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6820, 2437, 2339, 29} \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^2+\log (\log (13 x+34)) \]

[In]

Int[(13 + (68*x + 26*x^2)*Log[34 + 13*x])/((34 + 13*x)*Log[34 + 13*x]),x]

[Out]

x^2 + Log[Log[34 + 13*x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2339

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 2437

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \left (2 x+\frac {13}{(34+13 x) \log (34+13 x)}\right ) \, dx \\ & = x^2+13 \int \frac {1}{(34+13 x) \log (34+13 x)} \, dx \\ & = x^2+\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,34+13 x\right ) \\ & = x^2+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (34+13 x)\right ) \\ & = x^2+\log (\log (34+13 x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^2+\log (\log (34+13 x)) \]

[In]

Integrate[(13 + (68*x + 26*x^2)*Log[34 + 13*x])/((34 + 13*x)*Log[34 + 13*x]),x]

[Out]

x^2 + Log[Log[34 + 13*x]]

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

method	result	size
norman	\(x^{2}+\ln \left (\ln \left (13 x +34\right )\right )\)	\(12\)
risch	\(x^{2}+\ln \left (\ln \left (13 x +34\right )\right )\)	\(12\)
parts	\(x^{2}+\ln \left (\ln \left (13 x +34\right )\right )\)	\(12\)
parallelrisch	\(-\frac {1156}{169}+x^{2}+\ln \left (\ln \left (13 x +34\right )\right )\)	\(13\)
derivativedivides	\(\frac {\left (13 x +34\right )^{2}}{169}-\frac {68 x}{13}-\frac {2312}{169}+\ln \left (\ln \left (13 x +34\right )\right )\)	\(22\)
default	\(\frac {\left (13 x +34\right )^{2}}{169}-\frac {68 x}{13}-\frac {2312}{169}+\ln \left (\ln \left (13 x +34\right )\right )\)	\(22\)

[In]

int(((26*x^2+68*x)*ln(13*x+34)+13)/(13*x+34)/ln(13*x+34),x,method=_RETURNVERBOSE)

[Out]

x^2+ln(ln(13*x+34))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^{2} + \log \left (\log \left (13 \, x + 34\right )\right ) \]

[In]

integrate(((26*x^2+68*x)*log(13*x+34)+13)/(13*x+34)/log(13*x+34),x, algorithm="fricas")

[Out]

x^2 + log(log(13*x + 34))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^{2} + \log {\left (\log {\left (13 x + 34 \right )} \right )} \]

[In]

integrate(((26*x**2+68*x)*ln(13*x+34)+13)/(13*x+34)/ln(13*x+34),x)

[Out]

x**2 + log(log(13*x + 34))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^{2} + \log \left (\log \left (13 \, x + 34\right )\right ) \]

[In]

integrate(((26*x^2+68*x)*log(13*x+34)+13)/(13*x+34)/log(13*x+34),x, algorithm="maxima")

[Out]

x^2 + log(log(13*x + 34))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=x^{2} + \log \left (\log \left (13 \, x + 34\right )\right ) \]

[In]

integrate(((26*x^2+68*x)*log(13*x+34)+13)/(13*x+34)/log(13*x+34),x, algorithm="giac")

[Out]

x^2 + log(log(13*x + 34))

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {13+\left (68 x+26 x^2\right ) \log (34+13 x)}{(34+13 x) \log (34+13 x)} \, dx=\ln \left (\ln \left (13\,x+34\right )\right )+x^2 \]

[In]

int((log(13*x + 34)*(68*x + 26*x^2) + 13)/(log(13*x + 34)*(13*x + 34)),x)

[Out]

log(log(13*x + 34)) + x^2