3.85.99 \(\int \frac {12 x^2+4 x^3-6 x^4-12 x \log (16-48 x+36 x^2)+(-2+3 x) \log ^2(16-48 x+36 x^2)}{-2 x^2+3 x^3} \, dx\)

Optimal. Leaf size=31 \[ -x^2+\frac {x+x^2-\left (x-\log \left ((4-6 x)^2\right )\right )^2}{x} \]

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Rubi [A]  time = 0.41, antiderivative size = 54, normalized size of antiderivative = 1.74, number of steps used = 13, number of rules used = 11, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1593, 6742, 698, 2411, 2344, 2301, 2317, 2391, 2397, 2394, 2315} \begin {gather*} -x^2-\frac {3}{2} \log ^2\left (4 (2-3 x)^2\right )-\frac {(2-3 x) \log ^2\left (4 (2-3 x)^2\right )}{2 x}+4 \log (2-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12*x^2 + 4*x^3 - 6*x^4 - 12*x*Log[16 - 48*x + 36*x^2] + (-2 + 3*x)*Log[16 - 48*x + 36*x^2]^2)/(-2*x^2 + 3
*x^3),x]

[Out]

-x^2 + 4*Log[2 - 3*x] - (3*Log[4*(2 - 3*x)^2]^2)/2 - ((2 - 3*x)*Log[4*(2 - 3*x)^2]^2)/(2*x)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

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 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 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 2394

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

Rule 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 x^2+4 x^3-6 x^4-12 x \log \left (16-48 x+36 x^2\right )+(-2+3 x) \log ^2\left (16-48 x+36 x^2\right )}{x^2 (-2+3 x)} \, dx\\ &=\int \left (-\frac {2 \left (-6-2 x+3 x^2\right )}{-2+3 x}-\frac {12 \log \left (4 (2-3 x)^2\right )}{x (-2+3 x)}+\frac {\log ^2\left (4 (2-3 x)^2\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {-6-2 x+3 x^2}{-2+3 x} \, dx\right )-12 \int \frac {\log \left (4 (2-3 x)^2\right )}{x (-2+3 x)} \, dx+\int \frac {\log ^2\left (4 (2-3 x)^2\right )}{x^2} \, dx\\ &=-\frac {(2-3 x) \log ^2\left (4 (2-3 x)^2\right )}{2 x}-2 \int \left (x-\frac {6}{-2+3 x}\right ) \, dx-4 \operatorname {Subst}\left (\int \frac {\log \left (4 x^2\right )}{\left (\frac {2}{3}-\frac {x}{3}\right ) x} \, dx,x,2-3 x\right )-6 \int \frac {\log \left (4 (2-3 x)^2\right )}{x} \, dx\\ &=-x^2+4 \log (2-3 x)-\frac {(2-3 x) \log ^2\left (4 (2-3 x)^2\right )}{2 x}-6 \log \left (4 (2-3 x)^2\right ) \log \left (\frac {3 x}{2}\right )-2 \operatorname {Subst}\left (\int \frac {\log \left (4 x^2\right )}{\frac {2}{3}-\frac {x}{3}} \, dx,x,2-3 x\right )-6 \operatorname {Subst}\left (\int \frac {\log \left (4 x^2\right )}{x} \, dx,x,2-3 x\right )-36 \int \frac {\log \left (\frac {3 x}{2}\right )}{2-3 x} \, dx\\ &=-x^2+4 \log (2-3 x)-\frac {3}{2} \log ^2\left (4 (2-3 x)^2\right )-\frac {(2-3 x) \log ^2\left (4 (2-3 x)^2\right )}{2 x}-12 \text {Li}_2\left (1-\frac {3 x}{2}\right )-12 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,2-3 x\right )\\ &=-x^2+4 \log (2-3 x)-\frac {3}{2} \log ^2\left (4 (2-3 x)^2\right )-\frac {(2-3 x) \log ^2\left (4 (2-3 x)^2\right )}{2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 35, normalized size = 1.13 \begin {gather*} -x^2+2 \log \left (4 (2-3 x)^2\right )-\frac {\log ^2\left (4 (2-3 x)^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*x^2 + 4*x^3 - 6*x^4 - 12*x*Log[16 - 48*x + 36*x^2] + (-2 + 3*x)*Log[16 - 48*x + 36*x^2]^2)/(-2*x
^2 + 3*x^3),x]

[Out]

-x^2 + 2*Log[4*(2 - 3*x)^2] - Log[4*(2 - 3*x)^2]^2/x

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fricas [A]  time = 0.62, size = 36, normalized size = 1.16 \begin {gather*} -\frac {x^{3} - 2 \, x \log \left (36 \, x^{2} - 48 \, x + 16\right ) + \log \left (36 \, x^{2} - 48 \, x + 16\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-2)*log(36*x^2-48*x+16)^2-12*x*log(36*x^2-48*x+16)-6*x^4+4*x^3+12*x^2)/(3*x^3-2*x^2),x, algorit
hm="fricas")

[Out]

-(x^3 - 2*x*log(36*x^2 - 48*x + 16) + log(36*x^2 - 48*x + 16)^2)/x

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giac [A]  time = 0.22, size = 32, normalized size = 1.03 \begin {gather*} -x^{2} - \frac {\log \left (36 \, x^{2} - 48 \, x + 16\right )^{2}}{x} + 4 \, \log \left (3 \, x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-2)*log(36*x^2-48*x+16)^2-12*x*log(36*x^2-48*x+16)-6*x^4+4*x^3+12*x^2)/(3*x^3-2*x^2),x, algorit
hm="giac")

[Out]

-x^2 - log(36*x^2 - 48*x + 16)^2/x + 4*log(3*x - 2)

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maple [A]  time = 0.38, size = 33, normalized size = 1.06

method	result	size
risch	\(-\frac {\ln \left (36 x^{2}-48 x +16\right )^{2}}{x}-x^{2}+4 \ln \left (3 x -2\right )\)	\(33\)
norman	\(\frac {-x^{3}-\ln \left (36 x^{2}-48 x +16\right )^{2}}{x}+4 \ln \left (3 x -2\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x-2)*ln(36*x^2-48*x+16)^2-12*x*ln(36*x^2-48*x+16)-6*x^4+4*x^3+12*x^2)/(3*x^3-2*x^2),x,method=_RETURNVE
RBOSE)

[Out]

-1/x*ln(36*x^2-48*x+16)^2-x^2+4*ln(3*x-2)

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maxima [A]  time = 0.50, size = 42, normalized size = 1.35 \begin {gather*} -x^{2} - \frac {4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (3 \, x - 2\right ) + \log \left (3 \, x - 2\right )^{2}\right )}}{x} + 4 \, \log \left (3 \, x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-2)*log(36*x^2-48*x+16)^2-12*x*log(36*x^2-48*x+16)-6*x^4+4*x^3+12*x^2)/(3*x^3-2*x^2),x, algorit
hm="maxima")

[Out]

-x^2 - 4*(log(2)^2 + 2*log(2)*log(3*x - 2) + log(3*x - 2)^2)/x + 4*log(3*x - 2)

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mupad [B]  time = 0.18, size = 34, normalized size = 1.10 \begin {gather*} 2\,\ln \left ({\left (3\,x-2\right )}^2\right )-\frac {{\ln \left (36\,x^2-48\,x+16\right )}^2}{x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(36*x^2 - 48*x + 16)^2*(3*x - 2) - 12*x*log(36*x^2 - 48*x + 16) + 12*x^2 + 4*x^3 - 6*x^4)/(2*x^2 - 3*
x^3),x)

[Out]

2*log((3*x - 2)^2) - log(36*x^2 - 48*x + 16)^2/x - x^2

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sympy [A]  time = 0.17, size = 26, normalized size = 0.84 \begin {gather*} - x^{2} + 4 \log {\left (3 x - 2 \right )} - \frac {\log {\left (36 x^{2} - 48 x + 16 \right )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-2)*ln(36*x**2-48*x+16)**2-12*x*ln(36*x**2-48*x+16)-6*x**4+4*x**3+12*x**2)/(3*x**3-2*x**2),x)

[Out]

-x**2 + 4*log(3*x - 2) - log(36*x**2 - 48*x + 16)**2/x

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