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

Optimal. Leaf size=14 \[ x^2+\log (\log (-2+x+12 (3+x))) \]

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Rubi [A]  time = 0.11, antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6688, 2390, 2302, 29} \begin {gather*} x^2+\log (\log (13 x+34)) \end {gather*}

Antiderivative was successfully verified.

[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 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 (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+\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,34+13 x\right )\\ &=x^2+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (34+13 x)\right )\\ &=x^2+\log (\log (34+13 x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 11, normalized size = 0.79 \begin {gather*} x^2+\log (\log (34+13 x)) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.81, size = 11, normalized size = 0.79 \begin {gather*} x^{2} + \log \left (\log \left (13 \, x + 34\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.23, size = 11, normalized size = 0.79 \begin {gather*} x^{2} + \log \left (\log \left (13 \, x + 34\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.40, size = 12, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.72, size = 11, normalized size = 0.79 \begin {gather*} x^{2} + \log \left (\log \left (13 \, x + 34\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.81, size = 11, normalized size = 0.79 \begin {gather*} \ln \left (\ln \left (13\,x+34\right )\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.11, size = 10, normalized size = 0.71 \begin {gather*} x^{2} + \log {\left (\log {\left (13 x + 34 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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