3.89.6 \(\int \frac {(-144-72 x^2-6 x^3-9 x^4) \log (4)+(-64-20 x^2+6 x^3-3 x^4) \log ^2(4)}{144 x^2+72 x^4+9 x^6+(-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6) \log (4)+(256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6) \log ^2(4)} \, dx\) [8806]

Optimal. Leaf size=33 \[ \frac {x+\frac {1}{3} \left (3+\frac {x}{4+x^2}\right )}{-\frac {4}{3}-x+\frac {x}{\log (4)}} \]

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(33)=66\).
time = 0.30, antiderivative size = 98, normalized size of antiderivative = 2.97, number of steps used = 5, number of rules used = 3, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2099, 653, 209} \begin {gather*} \frac {\log (4) (3 (1-\log (4))-x \log (4))}{\left (x^2+4\right ) \left (9+13 \log ^2(4)-18 \log (4)\right )}+\frac {\log (4) \left (27+16 \log ^3(4)+15 \log ^2(4)-42 \log (4)\right )}{(1-\log (4)) \left (9+13 \log ^2(4)-18 \log (4)\right ) (3 x (1-\log (4))-\log (256))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-144 - 72*x^2 - 6*x^3 - 9*x^4)*Log[4] + (-64 - 20*x^2 + 6*x^3 - 3*x^4)*Log[4]^2)/(144*x^2 + 72*x^4 + 9*x
^6 + (-384*x - 288*x^2 - 192*x^3 - 144*x^4 - 24*x^5 - 18*x^6)*Log[4] + (256 + 384*x + 272*x^2 + 192*x^3 + 88*x
^4 + 24*x^5 + 9*x^6)*Log[4]^2),x]

[Out]

(Log[4]*(3*(1 - Log[4]) - x*Log[4]))/((4 + x^2)*(9 - 18*Log[4] + 13*Log[4]^2)) + (Log[4]*(27 - 42*Log[4] + 15*
Log[4]^2 + 16*Log[4]^3))/((1 - Log[4])*(9 - 18*Log[4] + 13*Log[4]^2)*(3*x*(1 - Log[4]) - Log[256]))

Rule 209

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

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 (-3 x (1-\log (4))-4 \log (4)) \log (4)}{\left (4+x^2\right )^2 \left (9-18 \log (4)+13 \log ^2(4)\right )}+\frac {\log ^2(4)}{\left (4+x^2\right ) \left (9-18 \log (4)+13 \log ^2(4)\right )}+\frac {3 \log (4) \left (-27+42 \log (4)-15 \log ^2(4)-16 \log ^3(4)\right )}{\left (9-18 \log (4)+13 \log ^2(4)\right ) (3 x (1-\log (4))-\log (256))^2}\right ) \, dx\\ &=\frac {\log (4) \left (27-42 \log (4)+15 \log ^2(4)+16 \log ^3(4)\right )}{(1-\log (4)) \left (9-18 \log (4)+13 \log ^2(4)\right ) (3 x (1-\log (4))-\log (256))}+\frac {(2 \log (4)) \int \frac {-3 x (1-\log (4))-4 \log (4)}{\left (4+x^2\right )^2} \, dx}{9-18 \log (4)+13 \log ^2(4)}+\frac {\log ^2(4) \int \frac {1}{4+x^2} \, dx}{9-18 \log (4)+13 \log ^2(4)}\\ &=\frac {\tan ^{-1}\left (\frac {x}{2}\right ) \log ^2(4)}{2 \left (9-18 \log (4)+13 \log ^2(4)\right )}+\frac {\log (4) (3 (1-\log (4))-x \log (4))}{\left (4+x^2\right ) \left (9-18 \log (4)+13 \log ^2(4)\right )}+\frac {\log (4) \left (27-42 \log (4)+15 \log ^2(4)+16 \log ^3(4)\right )}{(1-\log (4)) \left (9-18 \log (4)+13 \log ^2(4)\right ) (3 x (1-\log (4))-\log (256))}-\frac {\log ^2(4) \int \frac {1}{4+x^2} \, dx}{9-18 \log (4)+13 \log ^2(4)}\\ &=\frac {\log (4) (3 (1-\log (4))-x \log (4))}{\left (4+x^2\right ) \left (9-18 \log (4)+13 \log ^2(4)\right )}+\frac {\log (4) \left (27-42 \log (4)+15 \log ^2(4)+16 \log ^3(4)\right )}{(1-\log (4)) \left (9-18 \log (4)+13 \log ^2(4)\right ) (3 x (1-\log (4))-\log (256))}\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(346\) vs. \(2(33)=66\).
time = 0.26, size = 346, normalized size = 10.48 \begin {gather*} -\frac {\log (4) \left (3 x (-1+\log (4)) \left (1296+1296 \log ^4(4)+576 \log ^3(4) (-9+\log (256))-72 \log ^2(256)-19 \log ^4(256)-72 \log ^2(4) \left (-108+16 \log (256)+\log ^2(256)\right )+16 \log (4) \left (-324+36 \log (256)+9 \log ^2(256)+5 \log ^3(256)\right )\right )-3 \left (15552+1296 \log (256)+864 \log ^2(256)+168 \log ^3(256)+12 \log ^4(256)+\log ^5(256)+1296 \log ^4(4) (12+\log (256))-192 \log ^3(4) \left (324+27 \log (256)+2 \log ^2(256)\right )-48 \log (4) \left (1296+108 \log (256)+44 \log ^2(256)+7 \log ^3(256)\right )+24 \log ^2(4) \left (3888+324 \log (256)+68 \log ^2(256)+7 \log ^3(256)\right )\right )+x^2 \left (-11664+1296 \log ^5(4)-1296 \log (256)-648 \log ^2(256)-180 \log ^3(256)-9 \log ^4(256)-\log ^5(256)-1296 \log ^4(4) (13+\log (256))+72 \log ^3(4) \left (756+72 \log (256)+7 \log ^2(256)\right )-36 \log ^2(4) \left (2088+216 \log (256)+46 \log ^2(256)+5 \log ^3(256)\right )+\log (4) \left (47952+5184 \log (256)+1800 \log ^2(256)+360 \log ^3(256)+\log ^4(256)\right )\right )\right )}{3 \left (4+x^2\right ) (-1+\log (4)) (3 x (-1+\log (4))+\log (256)) \left (36-72 \log (4)+36 \log ^2(4)+\log ^2(256)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-144 - 72*x^2 - 6*x^3 - 9*x^4)*Log[4] + (-64 - 20*x^2 + 6*x^3 - 3*x^4)*Log[4]^2)/(144*x^2 + 72*x^4
+ 9*x^6 + (-384*x - 288*x^2 - 192*x^3 - 144*x^4 - 24*x^5 - 18*x^6)*Log[4] + (256 + 384*x + 272*x^2 + 192*x^3
+ 88*x^4 + 24*x^5 + 9*x^6)*Log[4]^2),x]

[Out]

-1/3*(Log[4]*(3*x*(-1 + Log[4])*(1296 + 1296*Log[4]^4 + 576*Log[4]^3*(-9 + Log[256]) - 72*Log[256]^2 - 19*Log[
256]^4 - 72*Log[4]^2*(-108 + 16*Log[256] + Log[256]^2) + 16*Log[4]*(-324 + 36*Log[256] + 9*Log[256]^2 + 5*Log[
256]^3)) - 3*(15552 + 1296*Log[256] + 864*Log[256]^2 + 168*Log[256]^3 + 12*Log[256]^4 + Log[256]^5 + 1296*Log[
4]^4*(12 + Log[256]) - 192*Log[4]^3*(324 + 27*Log[256] + 2*Log[256]^2) - 48*Log[4]*(1296 + 108*Log[256] + 44*L
og[256]^2 + 7*Log[256]^3) + 24*Log[4]^2*(3888 + 324*Log[256] + 68*Log[256]^2 + 7*Log[256]^3)) + x^2*(-11664 +
1296*Log[4]^5 - 1296*Log[256] - 648*Log[256]^2 - 180*Log[256]^3 - 9*Log[256]^4 - Log[256]^5 - 1296*Log[4]^4*(1
3 + Log[256]) + 72*Log[4]^3*(756 + 72*Log[256] + 7*Log[256]^2) - 36*Log[4]^2*(2088 + 216*Log[256] + 46*Log[256
]^2 + 5*Log[256]^3) + Log[4]*(47952 + 5184*Log[256] + 1800*Log[256]^2 + 360*Log[256]^3 + Log[256]^4))))/((4 +
x^2)*(-1 + Log[4])*(3*x*(-1 + Log[4]) + Log[256])*(36 - 72*Log[4] + 36*Log[4]^2 + Log[256]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(28)=56\).
time = 0.44, size = 97, normalized size = 2.94

method	result	size
norman	\(\frac {-\frac {3 \left (4 \ln \left (2\right )^{2}+6 \ln \left (2\right )\right ) x^{3}}{8 \ln \left (2\right )}-\frac {\left (16 \ln \left (2\right )^{2}+18 \ln \left (2\right )\right ) x}{2 \ln \left (2\right )}}{\left (x^{2}+4\right ) \left (6 x \ln \left (2\right )+8 \ln \left (2\right )-3 x \right )}\)	\(63\)
gosper	\(\frac {2 \left (2 x^{2} \ln \left (2\right )-2 x \ln \left (2\right )+3 x^{2}+8 \ln \left (2\right )+x +12\right ) \ln \left (2\right )}{\left (6 x^{3} \ln \left (2\right )+8 x^{2} \ln \left (2\right )-3 x^{3}+24 x \ln \left (2\right )+32 \ln \left (2\right )-12 x \right ) \left (2 \ln \left (2\right )-1\right )}\)	\(71\)
risch	\(\frac {\frac {\ln \left (2\right ) \left (2 \ln \left (2\right )+3\right ) x^{2}}{6 \ln \left (2\right )-3}-\frac {x \ln \left (2\right )}{3}+\frac {4 \ln \left (2\right ) \left (2 \ln \left (2\right )+3\right )}{3 \left (2 \ln \left (2\right )-1\right )}}{x^{3} \ln \left (2\right )+\frac {4 x^{2} \ln \left (2\right )}{3}-\frac {x^{3}}{2}+4 x \ln \left (2\right )+\frac {16 \ln \left (2\right )}{3}-2 x}\)	\(80\)
default	\(2 \ln \left (2\right ) \left (-\frac {-384 \ln \left (2\right )^{3}-180 \ln \left (2\right )^{2}+252 \ln \left (2\right )-81}{\left (52 \ln \left (2\right )^{2}-36 \ln \left (2\right )+9\right ) \left (6 \ln \left (2\right )-3\right ) \left (6 x \ln \left (2\right )+8 \ln \left (2\right )-3 x \right )}+\frac {-2 x \ln \left (2\right )+3-6 \ln \left (2\right )}{\left (x^{2}+4\right ) \left (52 \ln \left (2\right )^{2}-36 \ln \left (2\right )+9\right )}\right )\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(-3*x^4+6*x^3-20*x^2-64)*ln(2)^2+2*(-9*x^4-6*x^3-72*x^2-144)*ln(2))/(4*(9*x^6+24*x^5+88*x^4+192*x^3+272
*x^2+384*x+256)*ln(2)^2+2*(-18*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*ln(2)+9*x^6+72*x^4+144*x^2),x,method=
_RETURNVERBOSE)

[Out]

2*ln(2)*(-(-384*ln(2)^3-180*ln(2)^2+252*ln(2)-81)/(52*ln(2)^2-36*ln(2)+9)/(6*ln(2)-3)/(6*x*ln(2)+8*ln(2)-3*x)+
2/(52*ln(2)^2-36*ln(2)+9)*(-x*ln(2)+3/2-3*ln(2))/(x^2+4))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (30) = 60\).
time = 0.26, size = 103, normalized size = 3.12 \begin {gather*} \frac {2 \, {\left ({\left (2 \, \log \left (2\right )^{2} + 3 \, \log \left (2\right )\right )} x^{2} - {\left (2 \, \log \left (2\right )^{2} - \log \left (2\right )\right )} x + 8 \, \log \left (2\right )^{2} + 12 \, \log \left (2\right )\right )}}{3 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 1\right )} x^{3} + 8 \, {\left (2 \, \log \left (2\right )^{2} - \log \left (2\right )\right )} x^{2} + 12 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 1\right )} x + 64 \, \log \left (2\right )^{2} - 32 \, \log \left (2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144)*log(2))/(4*(9*x^6+24*x^5+88*x^4+192
*x^3+272*x^2+384*x+256)*log(2)^2+2*(-18*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2)
,x, algorithm="maxima")

[Out]

2*((2*log(2)^2 + 3*log(2))*x^2 - (2*log(2)^2 - log(2))*x + 8*log(2)^2 + 12*log(2))/(3*(4*log(2)^2 - 4*log(2) +
1)*x^3 + 8*(2*log(2)^2 - log(2))*x^2 + 12*(4*log(2)^2 - 4*log(2) + 1)*x + 64*log(2)^2 - 32*log(2))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).
time = 0.48, size = 79, normalized size = 2.39 \begin {gather*} \frac {2 \, {\left (2 \, {\left (x^{2} - x + 4\right )} \log \left (2\right )^{2} + {\left (3 \, x^{2} + x + 12\right )} \log \left (2\right )\right )}}{3 \, x^{3} + 4 \, {\left (3 \, x^{3} + 4 \, x^{2} + 12 \, x + 16\right )} \log \left (2\right )^{2} - 4 \, {\left (3 \, x^{3} + 2 \, x^{2} + 12 \, x + 8\right )} \log \left (2\right ) + 12 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144)*log(2))/(4*(9*x^6+24*x^5+88*x^4+192
*x^3+272*x^2+384*x+256)*log(2)^2+2*(-18*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2)
,x, algorithm="fricas")

[Out]

2*(2*(x^2 - x + 4)*log(2)^2 + (3*x^2 + x + 12)*log(2))/(3*x^3 + 4*(3*x^3 + 4*x^2 + 12*x + 16)*log(2)^2 - 4*(3*
x^3 + 2*x^2 + 12*x + 8)*log(2) + 12*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (22) = 44\).
time = 2.88, size = 102, normalized size = 3.09 \begin {gather*} - \frac {x^{2} \left (- 6 \log {\left (2 \right )} - 4 \log {\left (2 \right )}^{2}\right ) + x \left (- 2 \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2}\right ) - 24 \log {\left (2 \right )} - 16 \log {\left (2 \right )}^{2}}{x^{3} \left (- 12 \log {\left (2 \right )} + 3 + 12 \log {\left (2 \right )}^{2}\right ) + x^{2} \left (- 8 \log {\left (2 \right )} + 16 \log {\left (2 \right )}^{2}\right ) + x \left (- 48 \log {\left (2 \right )} + 12 + 48 \log {\left (2 \right )}^{2}\right ) - 32 \log {\left (2 \right )} + 64 \log {\left (2 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-3*x**4+6*x**3-20*x**2-64)*ln(2)**2+2*(-9*x**4-6*x**3-72*x**2-144)*ln(2))/(4*(9*x**6+24*x**5+88*
x**4+192*x**3+272*x**2+384*x+256)*ln(2)**2+2*(-18*x**6-24*x**5-144*x**4-192*x**3-288*x**2-384*x)*ln(2)+9*x**6+
72*x**4+144*x**2),x)

[Out]

-(x**2*(-6*log(2) - 4*log(2)**2) + x*(-2*log(2) + 4*log(2)**2) - 24*log(2) - 16*log(2)**2)/(x**3*(-12*log(2) +
3 + 12*log(2)**2) + x**2*(-8*log(2) + 16*log(2)**2) + x*(-48*log(2) + 12 + 48*log(2)**2) - 32*log(2) + 64*log
(2)**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).
time = 0.41, size = 82, normalized size = 2.48 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} \log \left (2\right )^{2} + 3 \, x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )^{2} + x \log \left (2\right ) + 8 \, \log \left (2\right )^{2} + 12 \, \log \left (2\right )\right )}}{{\left (6 \, x^{3} \log \left (2\right ) - 3 \, x^{3} + 8 \, x^{2} \log \left (2\right ) + 24 \, x \log \left (2\right ) - 12 \, x + 32 \, \log \left (2\right )\right )} {\left (2 \, \log \left (2\right ) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144)*log(2))/(4*(9*x^6+24*x^5+88*x^4+192
*x^3+272*x^2+384*x+256)*log(2)^2+2*(-18*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2)
,x, algorithm="giac")

[Out]

2*(2*x^2*log(2)^2 + 3*x^2*log(2) - 2*x*log(2)^2 + x*log(2) + 8*log(2)^2 + 12*log(2))/((6*x^3*log(2) - 3*x^3 +
8*x^2*log(2) + 24*x*log(2) - 12*x + 32*log(2))*(2*log(2) - 1))

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(2)^2*(20*x^2 - 6*x^3 + 3*x^4 + 64) + 2*log(2)*(72*x^2 + 6*x^3 + 9*x^4 + 144))/(4*log(2)^2*(384*x +
272*x^2 + 192*x^3 + 88*x^4 + 24*x^5 + 9*x^6 + 256) - 2*log(2)*(384*x + 288*x^2 + 192*x^3 + 144*x^4 + 24*x^5 +
18*x^6) + 144*x^2 + 72*x^4 + 9*x^6),x)

[Out]

\text{Hanged}

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