3.44.87 \(\int \frac {160-224 x+104 x^2-16 x^3+(-4-4 x+x^2) \log (4)}{448-288 x-80 x^2+72 x^3-8 x^4+(-4 x+x^3) \log (4)} \, dx\)

Optimal. Leaf size=25 \[ \log \left ((2+x) \left (-2 (7-x)-\frac {x \log (4)}{4 (-2+x)}\right )\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 28, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2074, 628} \begin {gather*} \log \left (8 x^2-x (72+\log (4))+112\right )-\log (2-x)+\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(160 - 224*x + 104*x^2 - 16*x^3 + (-4 - 4*x + x^2)*Log[4])/(448 - 288*x - 80*x^2 + 72*x^3 - 8*x^4 + (-4*x
+ x^3)*Log[4]),x]

[Out]

-Log[2 - x] + Log[2 + x] + Log[112 + 8*x^2 - x*(72 + Log[4])]

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 2074

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 {1}{2-x}+\frac {1}{2+x}+\frac {-72+16 x-\log (4)}{112+8 x^2-x (72+\log (4))}\right ) \, dx\\ &=-\log (2-x)+\log (2+x)+\int \frac {-72+16 x-\log (4)}{112+8 x^2+x (-72-\log (4))} \, dx\\ &=-\log (2-x)+\log (2+x)+\log \left (112+8 x^2-x (72+\log (4))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.16 \begin {gather*} -\log (2-x)+\log (2+x)+\log \left (112-72 x+8 x^2-x \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(160 - 224*x + 104*x^2 - 16*x^3 + (-4 - 4*x + x^2)*Log[4])/(448 - 288*x - 80*x^2 + 72*x^3 - 8*x^4 +
(-4*x + x^3)*Log[4]),x]

[Out]

-Log[2 - x] + Log[2 + x] + Log[112 - 72*x + 8*x^2 - x*Log[4]]

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fricas [A]  time = 0.52, size = 34, normalized size = 1.36 \begin {gather*} \log \left (4 \, x^{3} - 28 \, x^{2} - {\left (x^{2} + 2 \, x\right )} \log \relax (2) - 16 \, x + 112\right ) - \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(x^2-4*x-4)*log(2)-16*x^3+104*x^2-224*x+160)/(2*(x^3-4*x)*log(2)-8*x^4+72*x^3-80*x^2-288*x+448),x
, algorithm="fricas")

[Out]

log(4*x^3 - 28*x^2 - (x^2 + 2*x)*log(2) - 16*x + 112) - log(x - 2)

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giac [A]  time = 0.14, size = 30, normalized size = 1.20 \begin {gather*} \log \left ({\left | 4 \, x^{2} - x \log \relax (2) - 36 \, x + 56 \right |}\right ) + \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x - 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(x^2-4*x-4)*log(2)-16*x^3+104*x^2-224*x+160)/(2*(x^3-4*x)*log(2)-8*x^4+72*x^3-80*x^2-288*x+448),x
, algorithm="giac")

[Out]

log(abs(4*x^2 - x*log(2) - 36*x + 56)) + log(abs(x + 2)) - log(abs(x - 2))

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maple [A]  time = 0.08, size = 27, normalized size = 1.08

method	result	size
norman	\(-\ln \left (x -2\right )+\ln \left (2+x \right )+\ln \left (x \ln \relax (2)-4 x^{2}+36 x -56\right )\)	\(27\)
default	\(\ln \left (2+x \right )-\ln \left (x -2\right )+\ln \left (-x \ln \relax (2)+4 x^{2}-36 x +56\right )\)	\(28\)
risch	\(-\ln \left (x -2\right )+\ln \left (4 x^{3}+\left (-\ln \relax (2)-28\right ) x^{2}+\left (-2 \ln \relax (2)-16\right ) x +112\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(x^2-4*x-4)*ln(2)-16*x^3+104*x^2-224*x+160)/(2*(x^3-4*x)*ln(2)-8*x^4+72*x^3-80*x^2-288*x+448),x,method=
_RETURNVERBOSE)

[Out]

-ln(x-2)+ln(2+x)+ln(x*ln(2)-4*x^2+36*x-56)

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maxima [A]  time = 0.35, size = 26, normalized size = 1.04 \begin {gather*} \log \left (4 \, x^{2} - x {\left (\log \relax (2) + 36\right )} + 56\right ) + \log \left (x + 2\right ) - \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(x^2-4*x-4)*log(2)-16*x^3+104*x^2-224*x+160)/(2*(x^3-4*x)*log(2)-8*x^4+72*x^3-80*x^2-288*x+448),x
, algorithm="maxima")

[Out]

log(4*x^2 - x*(log(2) + 36) + 56) + log(x + 2) - log(x - 2)

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mupad [B]  time = 6.16, size = 4850, normalized size = 194.00 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((224*x + 2*log(2)*(4*x - x^2 + 4) - 104*x^2 + 16*x^3 - 160)/(288*x + 2*log(2)*(4*x - x^3) + 80*x^2 - 72*x^
3 + 8*x^4 - 448),x)

[Out]

symsum(log(2654208*log(2) - 132710400*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2
)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*l
og(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*lo
g(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 552
96*log(2)^5 - 256*log(2)^6, z, k) - 30228480*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^
4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 816742
4*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 81674
24*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^
4 - 55296*log(2)^5 - 256*log(2)^6, z, k)*log(2) + 66355200*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^
3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log
(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*lo
g(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4
083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)*x + 331776*x*log(2) - 1157120*root(530841600*z^4*log(2)
^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*lo
g(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z
*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^
2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)*log(2)^2 + 52494336*root(5308
41600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1
061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^
6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 5
30841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*log(2) - 19
456*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4
*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 51
2*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z
*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)*
log(2)^3 - 24920064*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*lo
g(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z
^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*
log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*
log(2)^6, z, k)^3*log(2) - 128*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 5
5296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4
- 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4
+ 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(
2)^5 - 256*log(2)^6, z, k)*log(2)^4 - 132710400*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712
*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 816
7424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 81
67424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(
2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*x + 66355200*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^
3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log
(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*lo
g(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4
083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^3*x + 183296*x*log(2)^2 + 6912*x*log(2)^3 + 64*x*log(2)
^4 + 265420800*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^
5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*lo
g(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2
)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2
)^6, z, k)^2 - 132710400*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z
^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110
592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 1105
92*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 -
256*log(2)^6, z, k)^3 + 136192*log(2)^2 + 5632*log(2)^3 + 64*log(2)^4 + 1905664*root(530841600*z^4*log(2)^2 +
110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)
^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log
(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 -
110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*log(2)^2 + 22016*root(530841600
*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 106168
3200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1
061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841
600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*log(2)^3 - 88473
6*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*l
og(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*
z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*l
og(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^3*
log(2)^2 + 64*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5
+ 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log
(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)
^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)
^6, z, k)^2*log(2)^4 - 8192*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 5529
6*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 -
110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 1
10592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^
5 - 256*log(2)^6, z, k)^3*log(2)^3 - 1204224*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^
4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 816742
4*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 81674
24*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^
4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*x*log(2)^2 - 20736*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(
2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*
log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z
*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3
- 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*x*log(2)^3 + 693760*root(530841600*z^4*log(2)^2 +
110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^
2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(
2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 1
10297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^3*x*log(2)^2 - 128*root(530841600*
z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683
200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 10
61683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 5308416
00*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*x*log(2)^4 + 1382
4*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*l
og(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*
z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*l
og(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^3*
x*log(2)^3 + 96*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)
^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*l
og(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(
2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(
2)^6, z, k)^3*x*log(2)^4 + 13455360*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^
4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log
(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(
2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296
*log(2)^5 - 256*log(2)^6, z, k)*x*log(2) + 327168*root(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 40837
12*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8
167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 +
8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*lo
g(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)*x*log(2)^2 - 27906048*root(530841600*z^4*log(2)^2 + 110297088*z^
4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*log(2)^2 - 22059417
6*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*z*log(2)^2 + 22059
4176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)^2 - 110297088*log
(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^2*x*log(2) + 14118912*root(530841600*z^4*log(2
)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061683200*z^3*l
og(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 + 1061683200*
z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 530841600*log(2)
^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)^3*x*log(2) - 32*root(5308416
00*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^6 - 1061
683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*log(2)^6 +
1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^6 - 5308
41600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k)*x*log(2)^4)*root
(530841600*z^4*log(2)^2 + 110297088*z^4*log(2)^3 + 4083712*z^4*log(2)^4 + 55296*z^4*log(2)^5 + 256*z^4*log(2)^
6 - 1061683200*z^3*log(2)^2 - 220594176*z^3*log(2)^3 - 8167424*z^3*log(2)^4 - 110592*z^3*log(2)^5 - 512*z^3*lo
g(2)^6 + 1061683200*z*log(2)^2 + 220594176*z*log(2)^3 + 8167424*z*log(2)^4 + 110592*z*log(2)^5 + 512*z*log(2)^
6 - 530841600*log(2)^2 - 110297088*log(2)^3 - 4083712*log(2)^4 - 55296*log(2)^5 - 256*log(2)^6, z, k), k, 1, 4
)

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sympy [A]  time = 1.19, size = 32, normalized size = 1.28 \begin {gather*} - \log {\left (x - 2 \right )} + \log {\left (x^{3} + x^{2} \left (-7 - \frac {\log {\relax (2 )}}{4}\right ) + x \left (-4 - \frac {\log {\relax (2 )}}{2}\right ) + 28 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(x**2-4*x-4)*ln(2)-16*x**3+104*x**2-224*x+160)/(2*(x**3-4*x)*ln(2)-8*x**4+72*x**3-80*x**2-288*x+4
48),x)

[Out]

-log(x - 2) + log(x**3 + x**2*(-7 - log(2)/4) + x*(-4 - log(2)/2) + 28)

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