3.84.35 \(\int \frac {(392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6) \log (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2})}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx\)
Optimal. Leaf size=30 \[ \log ^2\left (4 x \left ((7-x)^2-x+\frac {2}{2 x-x^3}\right )\right ) \]
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Rubi [F] time = 8.11, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((392 - 232*x - 368*x^2 + 240*x^3 + 74*x^4 - 60*x^5 + 6*x^6)*Log[(-8 - 392*x + 120*x^2 + 188*x^3 - 60*x^4
+ 4*x^5)/(-2 + x^2)])/(4 + 196*x - 62*x^2 - 192*x^3 + 60*x^4 + 45*x^5 - 15*x^6 + x^7),x]
[Out]
-Log[Sqrt[2] - x]^2 - 2*Log[(Sqrt[2] - x)/(2*Sqrt[2])]*Log[Sqrt[2] + x] - Log[Sqrt[2] + x]^2 - 2*Log[Sqrt[2] -
x]*Log[(Sqrt[2] + x)/(2*Sqrt[2])] - 2*Log[Sqrt[2] - x]*Log[(4*(2 + 98*x - 30*x^2 - 47*x^3 + 15*x^4 - x^5))/(2
- x^2)] - 2*Log[Sqrt[2] + x]*Log[(4*(2 + 98*x - 30*x^2 - 47*x^3 + 15*x^4 - x^5))/(2 - x^2)] - 2*PolyLog[2, (S
qrt[2] - x)/(2*Sqrt[2])] - 2*PolyLog[2, (Sqrt[2] + x)/(2*Sqrt[2])] - 196*Log[(4*(2 + 98*x - 30*x^2 - 47*x^3 +
15*x^4 - x^5))/(2 - x^2)]*Defer[Int][(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5)^(-1), x] + 120*Log[(4*(2 + 9
8*x - 30*x^2 - 47*x^3 + 15*x^4 - x^5))/(2 - x^2)]*Defer[Int][x/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x
] + 282*Log[(4*(2 + 98*x - 30*x^2 - 47*x^3 + 15*x^4 - x^5))/(2 - x^2)]*Defer[Int][x^2/(-2 - 98*x + 30*x^2 + 47
*x^3 - 15*x^4 + x^5), x] - 120*Log[(4*(2 + 98*x - 30*x^2 - 47*x^3 + 15*x^4 - x^5))/(2 - x^2)]*Defer[Int][x^3/(
-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 196*Defer[Int][Log[Sqrt[2] - x]/(-2 - 98*x + 30*x^2 + 47*x^3
- 15*x^4 + x^5), x] + 120*Defer[Int][(x*Log[Sqrt[2] - x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] +
282*Defer[Int][(x^2*Log[Sqrt[2] - x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 120*Defer[Int][(x^3*L
og[Sqrt[2] - x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 10*Defer[Int][(x^4*Log[Sqrt[2] - x])/(-2 -
98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 196*Defer[Int][Log[Sqrt[2] + x]/(-2 - 98*x + 30*x^2 + 47*x^3 - 1
5*x^4 + x^5), x] + 120*Defer[Int][(x*Log[Sqrt[2] + x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 282*
Defer[Int][(x^2*Log[Sqrt[2] + x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 120*Defer[Int][(x^3*Log[S
qrt[2] + x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 10*Defer[Int][(x^4*Log[Sqrt[2] + x])/(-2 - 98*
x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 10*Defer[Int][(x^4*Log[(4*(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^
5))/(-2 + x^2)])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 196*Defer[Int][Defer[Int][(-2 - 98*x + 30*
x^2 + 47*x^3 - 15*x^4 + x^5)^(-1), x]/(Sqrt[2] - x), x] - 196*Defer[Int][Defer[Int][(-2 - 98*x + 30*x^2 + 47*x
^3 - 15*x^4 + x^5)^(-1), x]/(Sqrt[2] + x), x] - 19208*Defer[Int][Defer[Int][(-2 - 98*x + 30*x^2 + 47*x^3 - 15*
x^4 + x^5)^(-1), x]/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 11760*Defer[Int][(x*Defer[Int][(-2 - 98
*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5)^(-1), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 27636*Defer[
Int][(x^2*Defer[Int][(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5)^(-1), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*
x^4 + x^5), x] - 11760*Defer[Int][(x^3*Defer[Int][(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5)^(-1), x])/(-2 -
98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 980*Defer[Int][(x^4*Defer[Int][(-2 - 98*x + 30*x^2 + 47*x^3 - 15
*x^4 + x^5)^(-1), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 120*Defer[Int][Defer[Int][x/(-2 - 98*
x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x]/(Sqrt[2] - x), x] + 120*Defer[Int][Defer[Int][x/(-2 - 98*x + 30*x^2 +
47*x^3 - 15*x^4 + x^5), x]/(Sqrt[2] + x), x] + 11760*Defer[Int][Defer[Int][x/(-2 - 98*x + 30*x^2 + 47*x^3 - 15
*x^4 + x^5), x]/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 7200*Defer[Int][(x*Defer[Int][x/(-2 - 98*x
+ 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 16920*Defer[Int][(x^
2*Defer[Int][x/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5),
x] + 7200*Defer[Int][(x^3*Defer[Int][x/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2
+ 47*x^3 - 15*x^4 + x^5), x] - 600*Defer[Int][(x^4*Defer[Int][x/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5),
x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 282*Defer[Int][Defer[Int][x^2/(-2 - 98*x + 30*x^2 + 47*
x^3 - 15*x^4 + x^5), x]/(Sqrt[2] - x), x] + 282*Defer[Int][Defer[Int][x^2/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^
4 + x^5), x]/(Sqrt[2] + x), x] + 27636*Defer[Int][Defer[Int][x^2/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5),
x]/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 16920*Defer[Int][(x*Defer[Int][x^2/(-2 - 98*x + 30*x^2
+ 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 39762*Defer[Int][(x^2*Defer[I
nt][x^2/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 1
6920*Defer[Int][(x^3*Defer[Int][x^2/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47
*x^3 - 15*x^4 + x^5), x] - 1410*Defer[Int][(x^4*Defer[Int][x^2/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x
])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 120*Defer[Int][Defer[Int][x^3/(-2 - 98*x + 30*x^2 + 47*x
^3 - 15*x^4 + x^5), x]/(Sqrt[2] - x), x] - 120*Defer[Int][Defer[Int][x^3/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4
+ x^5), x]/(Sqrt[2] + x), x] - 11760*Defer[Int][Defer[Int][x^3/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5),
x]/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 7200*Defer[Int][(x*Defer[Int][x^3/(-2 - 98*x + 30*x^2 +
47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] + 16920*Defer[Int][(x^2*Defer[Int
][x^3/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x] - 720
0*Defer[Int][(x^3*Defer[Int][x^3/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(-2 - 98*x + 30*x^2 + 47*x^
3 - 15*x^4 + x^5), x] + 600*Defer[Int][(x^4*Defer[Int][x^3/(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x])/(
-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5), x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 x \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2+x^2}+\frac {2 \left (-98+60 x+141 x^2-60 x^3+5 x^4\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5}\right ) \, dx\\ &=2 \int \frac {\left (-98+60 x+141 x^2-60 x^3+5 x^4\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5} \, dx-4 \int \frac {x \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2+x^2} \, dx\\ &=2 \int \left (-\frac {98 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5}+\frac {60 x \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5}+\frac {141 x^2 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5}-\frac {60 x^3 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5}+\frac {5 x^4 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5}\right ) \, dx-4 \int \left (-\frac {\log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{2 \left (\sqrt {2}-x\right )}+\frac {\log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{2 \left (\sqrt {2}+x\right )}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{\sqrt {2}-x} \, dx-2 \int \frac {\log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{\sqrt {2}+x} \, dx+10 \int \frac {x^4 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5} \, dx+120 \int \frac {x \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5} \, dx-120 \int \frac {x^3 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5} \, dx-196 \int \frac {\log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5} \, dx+282 \int \frac {x^2 \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{-2-98 x+30 x^2+47 x^3-15 x^4+x^5} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 35, normalized size = 1.17 \begin {gather*} \log ^2\left (\frac {4 \left (-2-98 x+30 x^2+47 x^3-15 x^4+x^5\right )}{-2+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((392 - 232*x - 368*x^2 + 240*x^3 + 74*x^4 - 60*x^5 + 6*x^6)*Log[(-8 - 392*x + 120*x^2 + 188*x^3 - 6
0*x^4 + 4*x^5)/(-2 + x^2)])/(4 + 196*x - 62*x^2 - 192*x^3 + 60*x^4 + 45*x^5 - 15*x^6 + x^7),x]
[Out]
Log[(4*(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5))/(-2 + x^2)]^2
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fricas [A] time = 0.57, size = 35, normalized size = 1.17 \begin {gather*} \log \left (\frac {4 \, {\left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )}}{x^{2} - 2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^4+188*x^3+120*x^2-392*x-8)/(x^2-2))/
(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x^2+196*x+4),x, algorithm="fricas")
[Out]
log(4*(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)/(x^2 - 2))^2
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giac [A] time = 0.16, size = 35, normalized size = 1.17 \begin {gather*} \log \left (\frac {4 \, {\left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )}}{x^{2} - 2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^4+188*x^3+120*x^2-392*x-8)/(x^2-2))/
(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x^2+196*x+4),x, algorithm="giac")
[Out]
log(4*(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)/(x^2 - 2))^2
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maple [A] time = 0.69, size = 37, normalized size = 1.23
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| method |
result |
size |
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| norman |
\(\ln \left (\frac {4 x^{5}-60 x^{4}+188 x^{3}+120 x^{2}-392 x -8}{x^{2}-2}\right )^{2}\) |
\(37\) |
| default |
\(\text {Expression too large to display}\) |
\(1544\) |
| risch |
\(\text {Expression too large to display}\) |
\(30279\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*ln((4*x^5-60*x^4+188*x^3+120*x^2-392*x-8)/(x^2-2))/(x^7-15
*x^6+45*x^5+60*x^4-192*x^3-62*x^2+196*x+4),x,method=_RETURNVERBOSE)
[Out]
ln((4*x^5-60*x^4+188*x^3+120*x^2-392*x-8)/(x^2-2))^2
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maxima [B] time = 0.38, size = 139, normalized size = 4.63 \begin {gather*} -\log \left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )^{2} + 2 \, \log \left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right ) \log \left (x^{2} - 2\right ) - \log \left (x^{2} - 2\right )^{2} + 2 \, {\left (\log \left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right ) - \log \left (x^{2} - 2\right )\right )} \log \left (\frac {4 \, {\left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )}}{x^{2} - 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^4+188*x^3+120*x^2-392*x-8)/(x^2-2))/
(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x^2+196*x+4),x, algorithm="maxima")
[Out]
-log(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)^2 + 2*log(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)*log(x^2 -
2) - log(x^2 - 2)^2 + 2*(log(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2) - log(x^2 - 2))*log(4*(x^5 - 15*x^4 +
47*x^3 + 30*x^2 - 98*x - 2)/(x^2 - 2))
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mupad [B] time = 6.04, size = 26, normalized size = 0.87 \begin {gather*} {\ln \left (196\,x-\frac {8}{x^2-2}-60\,x^2+4\,x^3\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((log(-(392*x - 120*x^2 - 188*x^3 + 60*x^4 - 4*x^5 + 8)/(x^2 - 2))*(240*x^3 - 368*x^2 - 232*x + 74*x^4 - 60
*x^5 + 6*x^6 + 392))/(196*x - 62*x^2 - 192*x^3 + 60*x^4 + 45*x^5 - 15*x^6 + x^7 + 4),x)
[Out]
log(196*x - 8/(x^2 - 2) - 60*x^2 + 4*x^3)^2
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sympy [A] time = 0.27, size = 32, normalized size = 1.07 \begin {gather*} \log {\left (\frac {4 x^{5} - 60 x^{4} + 188 x^{3} + 120 x^{2} - 392 x - 8}{x^{2} - 2} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((6*x**6-60*x**5+74*x**4+240*x**3-368*x**2-232*x+392)*ln((4*x**5-60*x**4+188*x**3+120*x**2-392*x-8)/(
x**2-2))/(x**7-15*x**6+45*x**5+60*x**4-192*x**3-62*x**2+196*x+4),x)
[Out]
log((4*x**5 - 60*x**4 + 188*x**3 + 120*x**2 - 392*x - 8)/(x**2 - 2))**2
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