3.52.52 \(\int \frac {21-8 x-11 x^2-2 x^3+(-32-46 x-16 x^2-2 x^3) \log (-\frac {\log (2)}{32+14 x+2 x^2})}{288-258 x-214 x^2+76 x^3+84 x^4+22 x^5+2 x^6} \, dx\)

Optimal. Leaf size=32 \[ \frac {\log \left (\frac {\log (2)}{2 \left (x-(4+x)^2\right )}\right )}{2 \left (-3+2 x+x^2\right )} \]

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Rubi [A]  time = 1.54, antiderivative size = 55, normalized size of antiderivative = 1.72, number of steps used = 44, number of rules used = 11, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6741, 12, 6728, 893, 634, 618, 204, 628, 2528, 2525, 800} \begin {gather*} -\frac {\log \left (-\frac {\log (2)}{2 \left (x^2+7 x+16\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (x^2+7 x+16\right )}\right )}{8 (x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(21 - 8*x - 11*x^2 - 2*x^3 + (-32 - 46*x - 16*x^2 - 2*x^3)*Log[-(Log[2]/(32 + 14*x + 2*x^2))])/(288 - 258*
x - 214*x^2 + 76*x^3 + 84*x^4 + 22*x^5 + 2*x^6),x]

[Out]

-1/8*Log[-1/2*Log[2]/(16 + 7*x + x^2)]/(1 - x) - Log[-1/2*Log[2]/(16 + 7*x + x^2)]/(8*(3 + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2525

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

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]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {21-8 x-11 x^2-2 x^3+\left (-32-46 x-16 x^2-2 x^3\right ) \log \left (-\frac {\log (2)}{32+14 x+2 x^2}\right )}{2 \left (3-2 x-x^2\right )^2 \left (16+7 x+x^2\right )} \, dx\\ &=\frac {1}{2} \int \frac {21-8 x-11 x^2-2 x^3+\left (-32-46 x-16 x^2-2 x^3\right ) \log \left (-\frac {\log (2)}{32+14 x+2 x^2}\right )}{\left (3-2 x-x^2\right )^2 \left (16+7 x+x^2\right )} \, dx\\ &=\frac {1}{2} \int \left (\frac {21}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )}-\frac {8 x}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )}-\frac {11 x^2}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )}-\frac {2 x^3}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )}-\frac {2 (1+x) \log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{(-1+x)^2 (3+x)^2}\right ) \, dx\\ &=-\left (4 \int \frac {x}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )} \, dx\right )-\frac {11}{2} \int \frac {x^2}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )} \, dx+\frac {21}{2} \int \frac {1}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )} \, dx-\int \frac {x^3}{(-1+x)^2 (3+x)^2 \left (16+7 x+x^2\right )} \, dx-\int \frac {(1+x) \log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{(-1+x)^2 (3+x)^2} \, dx\\ &=-\left (4 \int \left (\frac {1}{384 (-1+x)^2}+\frac {1}{3072 (-1+x)}-\frac {3}{64 (3+x)^2}+\frac {1}{256 (3+x)}+\frac {80-13 x}{3072 \left (16+7 x+x^2\right )}\right ) \, dx\right )-\frac {11}{2} \int \left (\frac {1}{384 (-1+x)^2}+\frac {3}{1024 (-1+x)}+\frac {9}{64 (3+x)^2}-\frac {15}{256 (3+x)}+\frac {208+171 x}{3072 \left (16+7 x+x^2\right )}\right ) \, dx+\frac {21}{2} \int \left (\frac {1}{384 (-1+x)^2}-\frac {7}{3072 (-1+x)}+\frac {1}{64 (3+x)^2}+\frac {1}{256 (3+x)}+\frac {-48-5 x}{3072 \left (16+7 x+x^2\right )}\right ) \, dx-\int \left (\frac {1}{384 (-1+x)^2}+\frac {17}{3072 (-1+x)}-\frac {27}{64 (3+x)^2}+\frac {81}{256 (3+x)}+\frac {-2736-989 x}{3072 \left (16+7 x+x^2\right )}\right ) \, dx-\int \left (\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (-1+x)^2}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)^2}\right ) \, dx\\ &=-\frac {3}{64} \log (1-x)+\frac {1}{32} \log (3+x)-\frac {\int \frac {-2736-989 x}{16+7 x+x^2} \, dx}{3072}-\frac {1}{768} \int \frac {80-13 x}{16+7 x+x^2} \, dx-\frac {11 \int \frac {208+171 x}{16+7 x+x^2} \, dx}{6144}+\frac {7 \int \frac {-48-5 x}{16+7 x+x^2} \, dx}{2048}-\frac {1}{8} \int \frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{(-1+x)^2} \, dx+\frac {1}{8} \int \frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{(3+x)^2} \, dx\\ &=-\frac {3}{64} \log (1-x)+\frac {1}{32} \log (3+x)-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)}+\frac {13 \int \frac {7+2 x}{16+7 x+x^2} \, dx}{1536}-\frac {35 \int \frac {7+2 x}{16+7 x+x^2} \, dx}{4096}-\frac {427 \int \frac {1}{16+7 x+x^2} \, dx}{4096}+\frac {1}{8} \int \frac {7+2 x}{(-1+x) \left (16+7 x+x^2\right )} \, dx-\frac {1}{8} \int \frac {7+2 x}{(3+x) \left (16+7 x+x^2\right )} \, dx-\frac {627 \int \frac {7+2 x}{16+7 x+x^2} \, dx}{4096}+\frac {989 \int \frac {7+2 x}{16+7 x+x^2} \, dx}{6144}-\frac {251 \int \frac {1}{16+7 x+x^2} \, dx}{1536}-\frac {1451 \int \frac {1}{16+7 x+x^2} \, dx}{6144}+\frac {8591 \int \frac {1}{16+7 x+x^2} \, dx}{12288}\\ &=-\frac {3}{64} \log (1-x)+\frac {1}{32} \log (3+x)+\frac {1}{128} \log \left (16+7 x+x^2\right )-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)}+\frac {1}{8} \int \left (\frac {3}{8 (-1+x)}+\frac {-8-3 x}{8 \left (16+7 x+x^2\right )}\right ) \, dx-\frac {1}{8} \int \left (\frac {1}{4 (3+x)}+\frac {4-x}{4 \left (16+7 x+x^2\right )}\right ) \, dx+\frac {427 \operatorname {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,7+2 x\right )}{2048}+\frac {251}{768} \operatorname {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,7+2 x\right )+\frac {1451 \operatorname {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,7+2 x\right )}{3072}-\frac {8591 \operatorname {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,7+2 x\right )}{6144}\\ &=\frac {5}{64} \sqrt {\frac {5}{3}} \tan ^{-1}\left (\frac {7+2 x}{\sqrt {15}}\right )+\frac {1}{128} \log \left (16+7 x+x^2\right )-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)}+\frac {1}{64} \int \frac {-8-3 x}{16+7 x+x^2} \, dx-\frac {1}{32} \int \frac {4-x}{16+7 x+x^2} \, dx\\ &=\frac {5}{64} \sqrt {\frac {5}{3}} \tan ^{-1}\left (\frac {7+2 x}{\sqrt {15}}\right )+\frac {1}{128} \log \left (16+7 x+x^2\right )-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)}+\frac {1}{64} \int \frac {7+2 x}{16+7 x+x^2} \, dx-\frac {3}{128} \int \frac {7+2 x}{16+7 x+x^2} \, dx+\frac {5}{128} \int \frac {1}{16+7 x+x^2} \, dx-\frac {15}{64} \int \frac {1}{16+7 x+x^2} \, dx\\ &=\frac {5}{64} \sqrt {\frac {5}{3}} \tan ^{-1}\left (\frac {7+2 x}{\sqrt {15}}\right )-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)}-\frac {5}{64} \operatorname {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,7+2 x\right )+\frac {15}{32} \operatorname {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,7+2 x\right )\\ &=-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (1-x)}-\frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{8 (3+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 31, normalized size = 0.97 \begin {gather*} \frac {\log \left (-\frac {\log (2)}{2 \left (16+7 x+x^2\right )}\right )}{2 \left (-3+2 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(21 - 8*x - 11*x^2 - 2*x^3 + (-32 - 46*x - 16*x^2 - 2*x^3)*Log[-(Log[2]/(32 + 14*x + 2*x^2))])/(288
- 258*x - 214*x^2 + 76*x^3 + 84*x^4 + 22*x^5 + 2*x^6),x]

[Out]

Log[-1/2*Log[2]/(16 + 7*x + x^2)]/(2*(-3 + 2*x + x^2))

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fricas [A]  time = 0.50, size = 27, normalized size = 0.84 \begin {gather*} \frac {\log \left (-\frac {\log \relax (2)}{2 \, {\left (x^{2} + 7 \, x + 16\right )}}\right )}{2 \, {\left (x^{2} + 2 \, x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-16*x^2-46*x-32)*log(-log(2)/(2*x^2+14*x+32))-2*x^3-11*x^2-8*x+21)/(2*x^6+22*x^5+84*x^4+76*x
^3-214*x^2-258*x+288),x, algorithm="fricas")

[Out]

1/2*log(-1/2*log(2)/(x^2 + 7*x + 16))/(x^2 + 2*x - 3)

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giac [A]  time = 0.24, size = 44, normalized size = 1.38 \begin {gather*} -\frac {\log \relax (2) - \log \left (-\log \relax (2)\right )}{2 \, {\left (x^{2} + 2 \, x - 3\right )}} - \frac {\log \left (x^{2} + 7 \, x + 16\right )}{2 \, {\left (x^{2} + 2 \, x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-16*x^2-46*x-32)*log(-log(2)/(2*x^2+14*x+32))-2*x^3-11*x^2-8*x+21)/(2*x^6+22*x^5+84*x^4+76*x
^3-214*x^2-258*x+288),x, algorithm="giac")

[Out]

-1/2*(log(2) - log(-log(2)))/(x^2 + 2*x - 3) - 1/2*log(x^2 + 7*x + 16)/(x^2 + 2*x - 3)

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maple [A]  time = 0.19, size = 30, normalized size = 0.94

method	result	size
norman	\(\frac {\ln \left (-\frac {\ln \relax (2)}{2 x^{2}+14 x +32}\right )}{2 x^{2}+4 x -6}\)	\(30\)
risch	\(\frac {\ln \left (-\frac {\ln \relax (2)}{2 x^{2}+14 x +32}\right )}{2 x^{2}+4 x -6}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-16*x^2-46*x-32)*ln(-ln(2)/(2*x^2+14*x+32))-2*x^3-11*x^2-8*x+21)/(2*x^6+22*x^5+84*x^4+76*x^3-214*x
^2-258*x+288),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(-ln(2)/(2*x^2+14*x+32))/(x^2+2*x-3)

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maxima [B]  time = 1.59, size = 122, normalized size = 3.81 \begin {gather*} -\frac {{\left (x^{2} + 2 \, x + 61\right )} \log \left (-x^{2} - 7 \, x - 16\right ) + 64 \, \log \relax (2) - 64 \, \log \left (\log \relax (2)\right )}{128 \, {\left (x^{2} + 2 \, x - 3\right )}} - \frac {161 \, x - 165}{384 \, {\left (x^{2} + 2 \, x - 3\right )}} + \frac {11 \, {\left (55 \, x - 51\right )}}{768 \, {\left (x^{2} + 2 \, x - 3\right )}} - \frac {17 \, x - 21}{96 \, {\left (x^{2} + 2 \, x - 3\right )}} - \frac {7 \, {\left (7 \, x - 3\right )}}{256 \, {\left (x^{2} + 2 \, x - 3\right )}} + \frac {1}{128} \, \log \left (x^{2} + 7 \, x + 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-16*x^2-46*x-32)*log(-log(2)/(2*x^2+14*x+32))-2*x^3-11*x^2-8*x+21)/(2*x^6+22*x^5+84*x^4+76*x
^3-214*x^2-258*x+288),x, algorithm="maxima")

[Out]

-1/128*((x^2 + 2*x + 61)*log(-x^2 - 7*x - 16) + 64*log(2) - 64*log(log(2)))/(x^2 + 2*x - 3) - 1/384*(161*x - 1
65)/(x^2 + 2*x - 3) + 11/768*(55*x - 51)/(x^2 + 2*x - 3) - 1/96*(17*x - 21)/(x^2 + 2*x - 3) - 7/256*(7*x - 3)/
(x^2 + 2*x - 3) + 1/128*log(x^2 + 7*x + 16)

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mupad [B]  time = 3.49, size = 34, normalized size = 1.06 \begin {gather*} \frac {\ln \left (\ln \relax (2)\right )-\ln \left (2\,x^2+14\,x+32\right )+\pi \,1{}\mathrm {i}}{2\,\left (x^2+2\,x-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + log(-log(2)/(14*x + 2*x^2 + 32))*(46*x + 16*x^2 + 2*x^3 + 32) + 11*x^2 + 2*x^3 - 21)/(76*x^3 - 214
*x^2 - 258*x + 84*x^4 + 22*x^5 + 2*x^6 + 288),x)

[Out]

(pi*1i + log(log(2)) - log(14*x + 2*x^2 + 32))/(2*(2*x + x^2 - 3))

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sympy [A]  time = 0.21, size = 26, normalized size = 0.81 \begin {gather*} \frac {\log {\left (- \frac {\log {\relax (2 )}}{2 x^{2} + 14 x + 32} \right )}}{2 x^{2} + 4 x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-16*x**2-46*x-32)*ln(-ln(2)/(2*x**2+14*x+32))-2*x**3-11*x**2-8*x+21)/(2*x**6+22*x**5+84*x**
4+76*x**3-214*x**2-258*x+288),x)

[Out]

log(-log(2)/(2*x**2 + 14*x + 32))/(2*x**2 + 4*x - 6)

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