3.90.10 \(\int \frac {(26+4 x) \log (-13 x-x^2)}{39 x+3 x^2} \, dx\)

Optimal. Leaf size=14 \[ \frac {1}{3} \log ^2((-13-x) x) \]

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Rubi [B]  time = 0.31, antiderivative size = 75, normalized size of antiderivative = 5.36, number of steps used = 18, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1593, 2528, 2524, 2357, 2301, 2317, 2391, 2418, 2392, 2390} \begin {gather*} \frac {2}{3} \log \left (-x^2-13 x\right ) \log (x)+\frac {2}{3} \log (x+13) \log \left (-x^2-13 x\right )-\frac {1}{3} \log ^2(x)-\frac {1}{3} \log ^2(x+13)-\frac {2}{3} \log \left (\frac {x}{13}+1\right ) \log (x)-\frac {2}{3} \log (13) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((26 + 4*x)*Log[-13*x - x^2])/(39*x + 3*x^2),x]

[Out]

(-2*Log[13]*Log[x])/3 - (2*Log[1 + x/13]*Log[x])/3 - Log[x]^2/3 - Log[13 + x]^2/3 + (2*Log[x]*Log[-13*x - x^2]
)/3 + (2*Log[13 + x]*Log[-13*x - x^2])/3

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

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 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2392

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(26+4 x) \log \left (-13 x-x^2\right )}{x (39+3 x)} \, dx\\ &=\int \left (\frac {2 \log \left (-13 x-x^2\right )}{3 x}+\frac {2 \log \left (-13 x-x^2\right )}{3 (13+x)}\right ) \, dx\\ &=\frac {2}{3} \int \frac {\log \left (-13 x-x^2\right )}{x} \, dx+\frac {2}{3} \int \frac {\log \left (-13 x-x^2\right )}{13+x} \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \frac {(-13-2 x) \log (x)}{-13 x-x^2} \, dx-\frac {2}{3} \int \frac {(-13-2 x) \log (13+x)}{-13 x-x^2} \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \frac {(-13-2 x) \log (x)}{(-13-x) x} \, dx-\frac {2}{3} \int \frac {(-13-2 x) \log (13+x)}{(-13-x) x} \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \left (\frac {\log (x)}{x}+\frac {\log (x)}{13+x}\right ) \, dx-\frac {2}{3} \int \left (\frac {\log (13+x)}{x}+\frac {\log (13+x)}{13+x}\right ) \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \frac {\log (x)}{x} \, dx-\frac {2}{3} \int \frac {\log (x)}{13+x} \, dx-\frac {2}{3} \int \frac {\log (13+x)}{x} \, dx-\frac {2}{3} \int \frac {\log (13+x)}{13+x} \, dx\\ &=-\frac {2}{3} \log (13) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{13}\right ) \log (x)-\frac {\log ^2(x)}{3}+\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,13+x\right )\\ &=-\frac {2}{3} \log (13) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{13}\right ) \log (x)-\frac {\log ^2(x)}{3}-\frac {1}{3} \log ^2(13+x)+\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 63, normalized size = 4.50 \begin {gather*} \frac {2}{3} \left (-\log (13) \log (x)-\frac {\log ^2(x)}{2}-\log (x) \log \left (\frac {13+x}{13}\right )-\frac {1}{2} \log ^2(13+x)+\log (x) \log (-x (13+x))+\log (13+x) \log (-x (13+x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((26 + 4*x)*Log[-13*x - x^2])/(39*x + 3*x^2),x]

[Out]

(2*(-(Log[13]*Log[x]) - Log[x]^2/2 - Log[x]*Log[(13 + x)/13] - Log[13 + x]^2/2 + Log[x]*Log[-(x*(13 + x))] + L
og[13 + x]*Log[-(x*(13 + x))]))/3

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fricas [A]  time = 0.55, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, \log \left (-x^{2} - 13 \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+26)*log(-x^2-13*x)/(3*x^2+39*x),x, algorithm="fricas")

[Out]

1/3*log(-x^2 - 13*x)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x + 13\right )} \log \left (-x^{2} - 13 \, x\right )}{3 \, {\left (x^{2} + 13 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+26)*log(-x^2-13*x)/(3*x^2+39*x),x, algorithm="giac")

[Out]

integrate(2/3*(2*x + 13)*log(-x^2 - 13*x)/(x^2 + 13*x), x)

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maple [A]  time = 0.44, size = 15, normalized size = 1.07

method	result	size
norman	\(\frac {\ln \left (-x^{2}-13 x \right )^{2}}{3}\)	\(15\)
risch	\(\frac {\ln \left (-x^{2}-13 x \right )^{2}}{3}\)	\(15\)
default	\(\frac {2 \ln \relax (x ) \ln \left (-x^{2}-13 x \right )}{3}-\frac {\ln \relax (x )^{2}}{3}-\frac {2 \ln \relax (x ) \ln \left (\frac {x}{13}+1\right )}{3}+\frac {2 \ln \left (x +13\right ) \ln \left (-x^{2}-13 x \right )}{3}-\frac {2 \left (\ln \left (x +13\right )-\ln \left (\frac {x}{13}+1\right )\right ) \ln \left (-\frac {x}{13}\right )}{3}-\frac {\ln \left (x +13\right )^{2}}{3}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x+26)*ln(-x^2-13*x)/(3*x^2+39*x),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(-x^2-13*x)^2

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maxima [B]  time = 0.36, size = 27, normalized size = 1.93 \begin {gather*} \frac {1}{3} \, \log \relax (x)^{2} + \frac {2}{3} \, \log \relax (x) \log \left (-x - 13\right ) + \frac {1}{3} \, \log \left (-x - 13\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+26)*log(-x^2-13*x)/(3*x^2+39*x),x, algorithm="maxima")

[Out]

1/3*log(x)^2 + 2/3*log(x)*log(-x - 13) + 1/3*log(-x - 13)^2

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mupad [B]  time = 6.71, size = 11, normalized size = 0.79 \begin {gather*} \frac {{\ln \left (-x\,\left (x+13\right )\right )}^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(- 13*x - x^2)*(4*x + 26))/(39*x + 3*x^2),x)

[Out]

log(-x*(x + 13))^2/3

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sympy [A]  time = 0.12, size = 12, normalized size = 0.86 \begin {gather*} \frac {\log {\left (- x^{2} - 13 x \right )}^{2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+26)*ln(-x**2-13*x)/(3*x**2+39*x),x)

[Out]

log(-x**2 - 13*x)**2/3

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