3.15.70 \(\int \frac {(-60-64 x) \log (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4})}{12+15 x+8 x^2} \, dx\)

Optimal. Leaf size=20 \[ \log ^2\left (\frac {1}{\left (\frac {5 x^2}{3}+(1+x) (4+x)\right )^2}\right ) \]

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Rubi [C]  time = 0.53, antiderivative size = 265, normalized size of antiderivative = 13.25, number of steps used = 20, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {2528, 2524, 2055, 2418, 2390, 2301, 2394, 2393, 2391} \begin {gather*} -8 \text {Li}_2\left (-\frac {16 i x-\sqrt {159}+15 i}{2 \sqrt {159}}\right )-8 \text {Li}_2\left (\frac {16 i x+\sqrt {159}+15 i}{2 \sqrt {159}}\right )-4 \log \left (\frac {9}{64 x^4+240 x^3+417 x^2+360 x+144}\right ) \log \left (16 x-i \sqrt {159}+15\right )-4 \log \left (16 x+i \sqrt {159}+15\right ) \log \left (\frac {9}{64 x^4+240 x^3+417 x^2+360 x+144}\right )-4 \log ^2\left (16 x-i \sqrt {159}+15\right )-4 \log ^2\left (16 x+i \sqrt {159}+15\right )-8 \log \left (-\frac {i \left (16 x+i \sqrt {159}+15\right )}{2 \sqrt {159}}\right ) \log \left (16 x-i \sqrt {159}+15\right )-8 \log \left (\frac {i \left (16 x-i \sqrt {159}+15\right )}{2 \sqrt {159}}\right ) \log \left (16 x+i \sqrt {159}+15\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-60 - 64*x)*Log[9/(144 + 360*x + 417*x^2 + 240*x^3 + 64*x^4)])/(12 + 15*x + 8*x^2),x]

[Out]

-4*Log[15 - I*Sqrt[159] + 16*x]^2 - 8*Log[((I/2)*(15 - I*Sqrt[159] + 16*x))/Sqrt[159]]*Log[15 + I*Sqrt[159] +
16*x] - 4*Log[15 + I*Sqrt[159] + 16*x]^2 - 8*Log[15 - I*Sqrt[159] + 16*x]*Log[((-1/2*I)*(15 + I*Sqrt[159] + 16
*x))/Sqrt[159]] - 4*Log[15 - I*Sqrt[159] + 16*x]*Log[9/(144 + 360*x + 417*x^2 + 240*x^3 + 64*x^4)] - 4*Log[15
+ I*Sqrt[159] + 16*x]*Log[9/(144 + 360*x + 417*x^2 + 240*x^3 + 64*x^4)] - 8*PolyLog[2, -1/2*(15*I - Sqrt[159]
+ (16*I)*x)/Sqrt[159]] - 8*PolyLog[2, (15*I + Sqrt[159] + (16*I)*x)/(2*Sqrt[159])]

Rule 2055

Int[(u_.)*(P_)*(Q_)^(q_), x_Symbol] :> Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(q + 1)*PolynomialQuotient[P
, gcd, x]*PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]] /; ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]

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 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 2393

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

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 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 \left (-\frac {64 \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15-i \sqrt {159}+16 x}-\frac {64 \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15+i \sqrt {159}+16 x}\right ) \, dx\\ &=-\left (64 \int \frac {\log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15-i \sqrt {159}+16 x} \, dx\right )-64 \int \frac {\log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15+i \sqrt {159}+16 x} \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \int \frac {\left (360+834 x+720 x^2+256 x^3\right ) \log \left (15-i \sqrt {159}+16 x\right )}{144+360 x+417 x^2+240 x^3+64 x^4} \, dx-4 \int \frac {\left (360+834 x+720 x^2+256 x^3\right ) \log \left (15+i \sqrt {159}+16 x\right )}{144+360 x+417 x^2+240 x^3+64 x^4} \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \int \frac {(30+32 x) \log \left (15-i \sqrt {159}+16 x\right )}{12+15 x+8 x^2} \, dx-4 \int \frac {(30+32 x) \log \left (15+i \sqrt {159}+16 x\right )}{12+15 x+8 x^2} \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \int \left (\frac {32 \log \left (15-i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x}+\frac {32 \log \left (15-i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x}\right ) \, dx-4 \int \left (\frac {32 \log \left (15+i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x}+\frac {32 \log \left (15+i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x}\right ) \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-128 \int \frac {\log \left (15-i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x} \, dx-128 \int \frac {\log \left (15-i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x} \, dx-128 \int \frac {\log \left (15+i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x} \, dx-128 \int \frac {\log \left (15+i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x} \, dx\\ &=-8 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )-8 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,15-i \sqrt {159}+16 x\right )-8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,15+i \sqrt {159}+16 x\right )+128 \int \frac {\log \left (\frac {16 \left (15-i \sqrt {159}+16 x\right )}{16 \left (15-i \sqrt {159}\right )-16 \left (15+i \sqrt {159}\right )}\right )}{15+i \sqrt {159}+16 x} \, dx+128 \int \frac {\log \left (\frac {16 \left (15+i \sqrt {159}+16 x\right )}{-16 \left (15-i \sqrt {159}\right )+16 \left (15+i \sqrt {159}\right )}\right )}{15-i \sqrt {159}+16 x} \, dx\\ &=-4 \log ^2\left (15-i \sqrt {159}+16 x\right )-8 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )-4 \log ^2\left (15+i \sqrt {159}+16 x\right )-8 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )+8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {16 x}{16 \left (15-i \sqrt {159}\right )-16 \left (15+i \sqrt {159}\right )}\right )}{x} \, dx,x,15+i \sqrt {159}+16 x\right )+8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {16 x}{-16 \left (15-i \sqrt {159}\right )+16 \left (15+i \sqrt {159}\right )}\right )}{x} \, dx,x,15-i \sqrt {159}+16 x\right )\\ &=-4 \log ^2\left (15-i \sqrt {159}+16 x\right )-8 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )-4 \log ^2\left (15+i \sqrt {159}+16 x\right )-8 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-8 \text {Li}_2\left (-\frac {15 i-\sqrt {159}+16 i x}{2 \sqrt {159}}\right )-8 \text {Li}_2\left (\frac {15 i+\sqrt {159}+16 i x}{2 \sqrt {159}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.07, size = 243, normalized size = 12.15 \begin {gather*} -4 \left (\log ^2\left (15-i \sqrt {159}+16 x\right )+2 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )+\log ^2\left (15+i \sqrt {159}+16 x\right )+2 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )+\log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{\left (12+15 x+8 x^2\right )^2}\right )+\log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{\left (12+15 x+8 x^2\right )^2}\right )+2 \text {Li}_2\left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )+2 \text {Li}_2\left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-60 - 64*x)*Log[9/(144 + 360*x + 417*x^2 + 240*x^3 + 64*x^4)])/(12 + 15*x + 8*x^2),x]

[Out]

-4*(Log[15 - I*Sqrt[159] + 16*x]^2 + 2*Log[((I/2)*(15 - I*Sqrt[159] + 16*x))/Sqrt[159]]*Log[15 + I*Sqrt[159] +
16*x] + Log[15 + I*Sqrt[159] + 16*x]^2 + 2*Log[15 - I*Sqrt[159] + 16*x]*Log[((-1/2*I)*(15 + I*Sqrt[159] + 16*
x))/Sqrt[159]] + Log[15 - I*Sqrt[159] + 16*x]*Log[9/(12 + 15*x + 8*x^2)^2] + Log[15 + I*Sqrt[159] + 16*x]*Log[
9/(12 + 15*x + 8*x^2)^2] + 2*PolyLog[2, ((I/2)*(15 - I*Sqrt[159] + 16*x))/Sqrt[159]] + 2*PolyLog[2, ((-1/2*I)*
(15 + I*Sqrt[159] + 16*x))/Sqrt[159]])

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fricas [A]  time = 0.77, size = 27, normalized size = 1.35 \begin {gather*} \log \left (\frac {9}{64 \, x^{4} + 240 \, x^{3} + 417 \, x^{2} + 360 \, x + 144}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-60)*log(9/(64*x^4+240*x^3+417*x^2+360*x+144))/(8*x^2+15*x+12),x, algorithm="fricas")

[Out]

log(9/(64*x^4 + 240*x^3 + 417*x^2 + 360*x + 144))^2

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giac [A]  time = 0.24, size = 27, normalized size = 1.35 \begin {gather*} \log \left (\frac {9}{64 \, x^{4} + 240 \, x^{3} + 417 \, x^{2} + 360 \, x + 144}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-60)*log(9/(64*x^4+240*x^3+417*x^2+360*x+144))/(8*x^2+15*x+12),x, algorithm="giac")

[Out]

log(9/(64*x^4 + 240*x^3 + 417*x^2 + 360*x + 144))^2

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maple [A]  time = 0.60, size = 28, normalized size = 1.40

method	result	size
norman	\(\ln \left (\frac {9}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )^{2}\)	\(28\)
default	\(-8 \ln \relax (3) \ln \left (8 x^{2}+15 x +12\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-8 \dilog \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right )^{2}-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-8 \dilog \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right )^{2}\)	\(215\)
risch	\(-8 \ln \relax (3) \ln \left (8 x^{2}+15 x +12\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-8 \dilog \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right )^{2}-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-8 \dilog \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right )^{2}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-64*x-60)*ln(9/(64*x^4+240*x^3+417*x^2+360*x+144))/(8*x^2+15*x+12),x,method=_RETURNVERBOSE)

[Out]

ln(9/(64*x^4+240*x^3+417*x^2+360*x+144))^2

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maxima [B]  time = 0.35, size = 54, normalized size = 2.70 \begin {gather*} -4 \, \log \left (8 \, x^{2} + 15 \, x + 12\right )^{2} - 4 \, \log \left (8 \, x^{2} + 15 \, x + 12\right ) \log \left (\frac {9}{64 \, x^{4} + 240 \, x^{3} + 417 \, x^{2} + 360 \, x + 144}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-60)*log(9/(64*x^4+240*x^3+417*x^2+360*x+144))/(8*x^2+15*x+12),x, algorithm="maxima")

[Out]

-4*log(8*x^2 + 15*x + 12)^2 - 4*log(8*x^2 + 15*x + 12)*log(9/(64*x^4 + 240*x^3 + 417*x^2 + 360*x + 144))

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mupad [B]  time = 0.15, size = 41, normalized size = 2.05 \begin {gather*} {\ln \left (64\,x^4+240\,x^3+417\,x^2+360\,x+144\right )}^2-4\,\ln \left ({\left (8\,x^2+15\,x+12\right )}^2\right )\,\ln \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(9/(360*x + 417*x^2 + 240*x^3 + 64*x^4 + 144))*(64*x + 60))/(15*x + 8*x^2 + 12),x)

[Out]

log(360*x + 417*x^2 + 240*x^3 + 64*x^4 + 144)^2 - 4*log((15*x + 8*x^2 + 12)^2)*log(3)

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sympy [A]  time = 0.26, size = 24, normalized size = 1.20 \begin {gather*} \log {\left (\frac {9}{64 x^{4} + 240 x^{3} + 417 x^{2} + 360 x + 144} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-60)*ln(9/(64*x**4+240*x**3+417*x**2+360*x+144))/(8*x**2+15*x+12),x)

[Out]

log(9/(64*x**4 + 240*x**3 + 417*x**2 + 360*x + 144))**2

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