3.2.22 \(\int \frac {1}{(a+8 x-8 x^2+4 x^3-x^4)^3} \, dx\) [122]
Optimal. Leaf size=252 \[ \frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {\left ((6+a) (25+7 a)+6 (7+2 a) (-1+x)^2\right ) (-1+x)}{32 (3+a)^2 (4+a)^2 \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {3 \left (80+7 a^2+14 \sqrt {4+a}+a \left (47+4 \sqrt {4+a}\right )\right ) \tan ^{-1}\left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^{5/2} \sqrt {1-\sqrt {4+a}}}-\frac {3 \left (14+4 a-\frac {80+47 a+7 a^2}{\sqrt {4+a}}\right ) \tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^2 \sqrt {1+\sqrt {4+a}}} \]
[Out]
1/8*(5+a+(-1+x)^2)*(-1+x)/(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-1+x)^4)^2+1/32*((6+a)*(25+7*a)+6*(7+2*a)*(-1+x)^2)*(-
1+x)/(a^2+7*a+12)^2/(3+a-2*(-1+x)^2-(-1+x)^4)-3/64*arctan((-1+x)/(1-(4+a)^(1/2))^(1/2))*(80+7*a^2+14*(4+a)^(1/
2)+a*(47+4*(4+a)^(1/2)))/(3+a)^2/(4+a)^(5/2)/(1-(4+a)^(1/2))^(1/2)-3/64*arctan((-1+x)/(1+(4+a)^(1/2))^(1/2))*(
14+4*a+(-7*a^2-47*a-80)/(4+a)^(1/2))/(3+a)^2/(4+a)^2/(1+(4+a)^(1/2))^(1/2)
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Rubi [A]
time = 0.45, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1120, 1106,
1192, 1180, 210} \begin {gather*} -\frac {3 \left (7 a^2+\left (4 \sqrt {a+4}+47\right ) a+14 \sqrt {a+4}+80\right ) \text {ArcTan}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt {1-\sqrt {a+4}}}-\frac {3 \left (-\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \text {ArcTan}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt {\sqrt {a+4}+1}}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-3),x]
[Out]
((5 + a + (-1 + x)^2)*(-1 + x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + (((6 + a)*(25 + 7
*a) + 6*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(32*(3 + a)^2*(4 + a)^2*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - (3*(80
+ 7*a^2 + 14*Sqrt[4 + a] + a*(47 + 4*Sqrt[4 + a]))*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]])/(64*(3 + a)^2*(4 +
a)^(5/2)*Sqrt[1 - Sqrt[4 + a]]) - (3*(14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqr
t[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt[1 + Sqrt[4 + a]])
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 1106
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*
x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p +
1)*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 -
4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Rule 1120
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - b*(d/(8*e)) + (c - 3*(d^2/(8*e
)))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]
Rule 1180
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rule 1192
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx &=\text {Subst}\left (\int \frac {1}{\left (3+a-2 x^2-x^4\right )^3} \, dx,x,-1+x\right )\\ &=\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}-\frac {\text {Subst}\left (\int \frac {4+2 (3+a)-4 (4+4 (3+a))-10 x^2}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )}{16 \left (12+7 a+a^2\right )}\\ &=-\frac {\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {\text {Subst}\left (\int \frac {12 \left (94+51 a+7 a^2\right )+24 (7+2 a) x^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )}{128 \left (12+7 a+a^2\right )^2}\\ &=-\frac {\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {\left (3 \left (14+4 a-\frac {80+47 a+7 a^2}{\sqrt {4+a}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{64 \left (12+7 a+a^2\right )^2}+\frac {\left (3 \left (80+7 a^2+14 \sqrt {4+a}+a \left (47+4 \sqrt {4+a}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-1+\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{64 \sqrt {4+a} \left (12+7 a+a^2\right )^2}\\ &=-\frac {\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {3 \left (80+47 a+7 a^2+\sqrt {4+a} (14+4 a)\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^{5/2} \sqrt {1-\sqrt {4+a}}}+\frac {3 \left (14+4 a-\frac {80+47 a+7 a^2}{\sqrt {4+a}}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{64 \left (12+7 a+a^2\right )^2 \sqrt {1+\sqrt {4+a}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.10, size = 254, normalized size = 1.01 \begin {gather*} \frac {1}{128} \left (\frac {16 (-1+x) \left (6+a-2 x+x^2\right )}{(3+a) (4+a) \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )^2}+\frac {4 (-1+x) \left (7 a^2+6 \left (32-14 x+7 x^2\right )+a \left (79-24 x+12 x^2\right )\right )}{(3+a)^2 (4+a)^2 \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )}-\frac {3 \text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {108 \log (x-\text {$\#$1})+55 a \log (x-\text {$\#$1})+7 a^2 \log (x-\text {$\#$1})-28 \log (x-\text {$\#$1}) \text {$\#$1}-8 a \log (x-\text {$\#$1}) \text {$\#$1}+14 \log (x-\text {$\#$1}) \text {$\#$1}^2+4 a \log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]}{\left (12+7 a+a^2\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-3),x]
[Out]
((16*(-1 + x)*(6 + a - 2*x + x^2))/((3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^2) + (4*(-1 + x)*(7*a^2 +
6*(32 - 14*x + 7*x^2) + a*(79 - 24*x + 12*x^2)))/((3 + a)^2*(4 + a)^2*(a - x*(-8 + 8*x - 4*x^2 + x^3))) - (3*
RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (108*Log[x - #1] + 55*a*Log[x - #1] + 7*a^2*Log[x - #1] - 28*Log
[x - #1]*#1 - 8*a*Log[x - #1]*#1 + 14*Log[x - #1]*#1^2 + 4*a*Log[x - #1]*#1^2)/(-2 + 4*#1 - 3*#1^2 + #1^3) & ]
)/(12 + 7*a + a^2)^2)/128
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.05, size = 400, normalized size = 1.59
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {\frac {3 \left (7+2 a \right ) x^{7}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {21 \left (7+2 a \right ) x^{6}}{16 \left (a^{2}+8 a +16\right ) \left (a^{2}+6 a +9\right )}+\frac {\left (7 a^{2}+343 a +1116\right ) x^{5}}{32 a^{4}+448 a^{3}+2336 a^{2}+5376 a +4608}-\frac {5 \left (7 a^{2}+175 a +528\right ) x^{4}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (34 a^{2}+679 a +1968\right ) x^{3}}{16 a^{4}+224 a^{3}+1168 a^{2}+2688 a +2304}-\frac {\left (32 a^{2}+623 a +1800\right ) x^{2}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {\left (11 a^{3}+107 a^{2}-84 a -1152\right ) x}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {11 a^{3}+131 a^{2}+408 a +288}{32 \left (3+a \right ) \left (a^{3}+11 a^{2}+40 a +48\right )}}{\left (-x^{4}+4 x^{3}-8 x^{2}+a +8 x \right )^{2}}-\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (-108+2 \left (-2 a -7\right ) \textit {\_R}^{2}+4 \left (7+2 a \right ) \textit {\_R} -7 a^{2}-55 a \right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{128 \left (a^{3}+10 a^{2}+33 a +36\right ) \left (4+a \right )}\) |
\(400\) |
| risch |
\(\frac {-\frac {3 \left (7+2 a \right ) x^{7}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {21 \left (7+2 a \right ) x^{6}}{16 \left (a^{2}+8 a +16\right ) \left (a^{2}+6 a +9\right )}-\frac {\left (7 a^{2}+343 a +1116\right ) x^{5}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {5 \left (7 a^{2}+175 a +528\right ) x^{4}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {\left (34 a^{2}+679 a +1968\right ) x^{3}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (32 a^{2}+623 a +1800\right ) x^{2}}{16 a^{4}+224 a^{3}+1168 a^{2}+2688 a +2304}+\frac {\left (11 a^{3}+107 a^{2}-84 a -1152\right ) x}{32 a^{4}+448 a^{3}+2336 a^{2}+5376 a +4608}-\frac {11 a^{3}+131 a^{2}+408 a +288}{32 \left (3+a \right ) \left (a^{3}+11 a^{2}+40 a +48\right )}}{\left (-x^{4}+4 x^{3}-8 x^{2}+a +8 x \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {2 \left (7+2 a \right ) \textit {\_R}^{2}}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}-\frac {4 \left (7+2 a \right ) \textit {\_R}}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}+\frac {7 a +27}{a^{3}+10 a^{2}+33 a +36}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{128}\) |
\(431\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-x^4+4*x^3-8*x^2+a+8*x)^3,x,method=_RETURNVERBOSE)
[Out]
-(3/16*(7+2*a)/(a^4+14*a^3+73*a^2+168*a+144)*x^7-21/16*(7+2*a)/(a^2+8*a+16)/(a^2+6*a+9)*x^6+1/32*(7*a^2+343*a+
1116)/(a^4+14*a^3+73*a^2+168*a+144)*x^5-5/32*(7*a^2+175*a+528)/(a^4+14*a^3+73*a^2+168*a+144)*x^4+1/16*(34*a^2+
679*a+1968)/(a^4+14*a^3+73*a^2+168*a+144)*x^3-1/16*(32*a^2+623*a+1800)/(a^4+14*a^3+73*a^2+168*a+144)*x^2-1/32*
(11*a^3+107*a^2-84*a-1152)/(a^4+14*a^3+73*a^2+168*a+144)*x+1/32*(11*a^3+131*a^2+408*a+288)/(3+a)/(a^3+11*a^2+4
0*a+48))/(-x^4+4*x^3-8*x^2+a+8*x)^2-3/128/(a^3+10*a^2+33*a+36)/(4+a)*sum((-108+2*(-2*a-7)*_R^2+4*(7+2*a)*_R-7*
a^2-55*a)/(-_R^3+3*_R^2-4*_R+2)*ln(x-_R),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a))
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="maxima")
[Out]
-1/32*(6*(2*a + 7)*x^7 - 42*(2*a + 7)*x^6 + (7*a^2 + 343*a + 1116)*x^5 - 5*(7*a^2 + 175*a + 528)*x^4 + 2*(34*a
^2 + 679*a + 1968)*x^3 + 11*a^3 - 2*(32*a^2 + 623*a + 1800)*x^2 + 131*a^2 - (11*a^3 + 107*a^2 - 84*a - 1152)*x
+ 408*a + 288)/((a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*
(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^6 + a^6 - 80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a
^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a
- 2304)*x^3 + 168*a^3 - 16*(a^5 + 10*a^4 + 17*a^3 - 124*a^2 - 528*a - 576)*x^2 + 144*a^2 + 16*(a^5 + 14*a^4 +
73*a^3 + 168*a^2 + 144*a)*x) - 3/32*integrate((2*(2*a + 7)*x^2 + 7*a^2 - 4*(2*a + 7)*x + 55*a + 108)/(x^4 - 4
*x^3 + 8*x^2 - a - 8*x), x)/(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3971 vs.
\(2 (220) = 440\).
time = 0.42, size = 3971, normalized size = 15.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="fricas")
[Out]
-1/128*(24*(2*a + 7)*x^7 - 168*(2*a + 7)*x^6 + 4*(7*a^2 + 343*a + 1116)*x^5 - 20*(7*a^2 + 175*a + 528)*x^4 + 8
*(34*a^2 + 679*a + 1968)*x^3 + 44*a^3 - 8*(32*a^2 + 623*a + 1800)*x^2 - 3*((a^4 + 14*a^3 + 73*a^2 + 168*a + 14
4)*x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^6 + a^6 - 80
*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216)*x
^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*x^3 + 168*a^3 - 16*(a^5 + 10*a^4 + 17*a^3 -
124*a^2 - 528*a - 576)*x^2 + 144*a^2 + 16*(a^5 + 14*a^4 + 73*a^3 + 168*a^2 + 144*a)*x)*sqrt((105*a^4 + 1470*a
^3 + 7749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 95040
0*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165
*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 +
477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 18228*a + 16144)/(a
^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a +
248832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^3 + 177061*a^2 + 415884*a + 36753
6)*x + 27*(343*a^7 + 8981*a^6 + 100811*a^5 + 628887*a^4 + 2354874*a^3 + 5293208*a^2 - (11*a^12 + 462*a^11 + 88
81*a^10 + 103320*a^9 + 810205*a^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*a^4 + 181238400*a^3
+ 187875072*a^2 + 117863424*a + 33841152)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 +
50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 2
41870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 6613
472*a + 3543424)*sqrt((105*a^4 + 1470*a^3 + 7749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 12936
7*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 39
8164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 2857032
0*a^8 + 94320045*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 2771366
40*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4
+ 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 9923472) + 3*((a^4 + 14*a^3 + 73*a^2 + 168*a +
144)*x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^6 + a^6 -
80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216
)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*x^3 + 168*a^3 - 16*(a^5 + 10*a^4 + 17*a^
3 - 124*a^2 - 528*a - 576)*x^2 + 144*a^2 + 16*(a^5 + 14*a^4 + 73*a^3 + 168*a^2 + 144*a)*x)*sqrt((105*a^4 + 147
0*a^3 + 7749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 95
0400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1
165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6
+ 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 18228*a + 16144)
/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*
a + 248832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^3 + 177061*a^2 + 415884*a + 36
7536)*x - 27*(343*a^7 + 8981*a^6 + 100811*a^5 + 628887*a^4 + 2354874*a^3 + 5293208*a^2 - (11*a^12 + 462*a^11 +
8881*a^10 + 103320*a^9 + 810205*a^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*a^4 + 181238400*a
^3 + 187875072*a^2 + 117863424*a + 33841152)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^1
5 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7
+ 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 6
613472*a + 3543424)*sqrt((105*a^4 + 1470*a^3 + 7749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 12
9367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 +
398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 2857
0320*a^8 + 94320045*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 2771
36640*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*
a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 9923472) - 3*((a^4 + 14*a^3 + 73*a^2 + 168*
a + 144)*x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^6 + a^
6 - 80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^...
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 697 vs.
\(2 (230) = 460\).
time = 9.46, size = 697, normalized size = 2.77 \begin {gather*} - \frac {11 a^{3} + 131 a^{2} + 408 a + x^{7} \cdot \left (12 a + 42\right ) + x^{6} \left (- 84 a - 294\right ) + x^{5} \cdot \left (7 a^{2} + 343 a + 1116\right ) + x^{4} \left (- 35 a^{2} - 875 a - 2640\right ) + x^{3} \cdot \left (68 a^{2} + 1358 a + 3936\right ) + x^{2} \left (- 64 a^{2} - 1246 a - 3600\right ) + x \left (- 11 a^{3} - 107 a^{2} + 84 a + 1152\right ) + 288}{32 a^{6} + 448 a^{5} + 2336 a^{4} + 5376 a^{3} + 4608 a^{2} + x^{8} \cdot \left (32 a^{4} + 448 a^{3} + 2336 a^{2} + 5376 a + 4608\right ) + x^{7} \left (- 256 a^{4} - 3584 a^{3} - 18688 a^{2} - 43008 a - 36864\right ) + x^{6} \cdot \left (1024 a^{4} + 14336 a^{3} + 74752 a^{2} + 172032 a + 147456\right ) + x^{5} \left (- 2560 a^{4} - 35840 a^{3} - 186880 a^{2} - 430080 a - 368640\right ) + x^{4} \left (- 64 a^{5} + 3200 a^{4} + 52672 a^{3} + 288256 a^{2} + 678912 a + 589824\right ) + x^{3} \cdot \left (256 a^{5} - 512 a^{4} - 38656 a^{3} - 256000 a^{2} - 651264 a - 589824\right ) + x^{2} \left (- 512 a^{5} - 5120 a^{4} - 8704 a^{3} + 63488 a^{2} + 270336 a + 294912\right ) + x \left (512 a^{5} + 7168 a^{4} + 37376 a^{3} + 86016 a^{2} + 73728 a\right )} - \operatorname {RootSum} {\left (t^{4} \cdot \left (268435456 a^{15} + 14763950080 a^{14} + 378493992960 a^{13} + 5999532441600 a^{12} + 65757291479040 a^{11} + 527875908304896 a^{10} + 3206246773555200 a^{9} + 15003759578972160 a^{8} + 54537151127224320 a^{7} + 153980418717122560 a^{6} + 334927734494986240 a^{5} + 551152193655275520 a^{4} + 664192984106926080 a^{3} + 553362212027105280 a^{2} + 284993413919539200 a + 68398419340689408\right ) + t^{2} \left (- 30965760 a^{9} - 1052835840 a^{8} - 15910207488 a^{7} - 140262506496 a^{6} - 795007254528 a^{5} - 3004516270080 a^{4} - 7571263979520 a^{3} - 12268037210112 a^{2} - 11598827618304 a - 4875324751872\right ) - 194481 a^{4} - 2762424 a^{3} - 14762736 a^{2} - 35178624 a - 31539456, \left ( t \mapsto t \log {\left (x + \frac {23068672 t^{3} a^{12} + 968884224 t^{3} a^{11} + 18624806912 t^{3} a^{10} + 216677744640 t^{3} a^{9} + 1699123036160 t^{3} a^{8} + 9461389328384 t^{3} a^{7} + 38361186172928 t^{3} a^{6} + 114107491549184 t^{3} a^{5} + 247138458009600 t^{3} a^{4} + 380084473036800 t^{3} a^{3} + 394002582994944 t^{3} a^{2} + 247177515368448 t^{3} a + 70970039599104 t^{3} - 395136 t a^{7} - 11676672 t a^{6} - 144076032 t a^{5} - 969518592 t a^{4} - 3861475200 t a^{3} - 9133300224 t a^{2} - 11906574336 t a - 6611337216 t - 64827 a^{4} - 907578 a^{3} - 4780647 a^{2} - 11228868 a - 9923472}{64827 a^{4} + 907578 a^{3} + 4780647 a^{2} + 11228868 a + 9923472} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x**4+4*x**3-8*x**2+a+8*x)**3,x)
[Out]
-(11*a**3 + 131*a**2 + 408*a + x**7*(12*a + 42) + x**6*(-84*a - 294) + x**5*(7*a**2 + 343*a + 1116) + x**4*(-3
5*a**2 - 875*a - 2640) + x**3*(68*a**2 + 1358*a + 3936) + x**2*(-64*a**2 - 1246*a - 3600) + x*(-11*a**3 - 107*
a**2 + 84*a + 1152) + 288)/(32*a**6 + 448*a**5 + 2336*a**4 + 5376*a**3 + 4608*a**2 + x**8*(32*a**4 + 448*a**3
+ 2336*a**2 + 5376*a + 4608) + x**7*(-256*a**4 - 3584*a**3 - 18688*a**2 - 43008*a - 36864) + x**6*(1024*a**4 +
14336*a**3 + 74752*a**2 + 172032*a + 147456) + x**5*(-2560*a**4 - 35840*a**3 - 186880*a**2 - 430080*a - 36864
0) + x**4*(-64*a**5 + 3200*a**4 + 52672*a**3 + 288256*a**2 + 678912*a + 589824) + x**3*(256*a**5 - 512*a**4 -
38656*a**3 - 256000*a**2 - 651264*a - 589824) + x**2*(-512*a**5 - 5120*a**4 - 8704*a**3 + 63488*a**2 + 270336*
a + 294912) + x*(512*a**5 + 7168*a**4 + 37376*a**3 + 86016*a**2 + 73728*a)) - RootSum(_t**4*(268435456*a**15 +
14763950080*a**14 + 378493992960*a**13 + 5999532441600*a**12 + 65757291479040*a**11 + 527875908304896*a**10 +
3206246773555200*a**9 + 15003759578972160*a**8 + 54537151127224320*a**7 + 153980418717122560*a**6 + 334927734
494986240*a**5 + 551152193655275520*a**4 + 664192984106926080*a**3 + 553362212027105280*a**2 + 284993413919539
200*a + 68398419340689408) + _t**2*(-30965760*a**9 - 1052835840*a**8 - 15910207488*a**7 - 140262506496*a**6 -
795007254528*a**5 - 3004516270080*a**4 - 7571263979520*a**3 - 12268037210112*a**2 - 11598827618304*a - 4875324
751872) - 194481*a**4 - 2762424*a**3 - 14762736*a**2 - 35178624*a - 31539456, Lambda(_t, _t*log(x + (23068672*
_t**3*a**12 + 968884224*_t**3*a**11 + 18624806912*_t**3*a**10 + 216677744640*_t**3*a**9 + 1699123036160*_t**3*
a**8 + 9461389328384*_t**3*a**7 + 38361186172928*_t**3*a**6 + 114107491549184*_t**3*a**5 + 247138458009600*_t*
*3*a**4 + 380084473036800*_t**3*a**3 + 394002582994944*_t**3*a**2 + 247177515368448*_t**3*a + 70970039599104*_
t**3 - 395136*_t*a**7 - 11676672*_t*a**6 - 144076032*_t*a**5 - 969518592*_t*a**4 - 3861475200*_t*a**3 - 913330
0224*_t*a**2 - 11906574336*_t*a - 6611337216*_t - 64827*a**4 - 907578*a**3 - 4780647*a**2 - 11228868*a - 99234
72)/(64827*a**4 + 907578*a**3 + 4780647*a**2 + 11228868*a + 9923472))))
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [sageVARa]=[86]Warning
, need to choose a br
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Mupad [B]
time = 6.41, size = 2500, normalized size = 9.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x)
[Out]
atan(((((52357496832*a + 57139003392*a^2 + 36322148352*a^3 + 14822473728*a^4 + 4027170816*a^5 + 728506368*a^6
+ 84615168*a^7 + 5726208*a^8 + 172032*a^9 + 21290287104)/(16384*(940032*a + 1195776*a^2 + 899328*a^3 + 442864*
a^4 + 149208*a^5 + 34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) + ((4290672328704*a + 60011430543
36*a^2 + 5025917042688*a^3 + 2800520003584*a^4 + 1090200272896*a^5 + 302556119040*a^6 + 59862155264*a^7 + 8275
361792*a^8 + 761266176*a^9 + 41943040*a^10 + 1048576*a^11 + 1391569403904)/(16384*(940032*a + 1195776*a^2 + 89
9328*a^3 + 442864*a^4 + 149208*a^5 + 34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) - (x*(351063244
8*a + 4020240384*a^2 + 2678587392*a^3 + 1144324096*a^4 + 325074944*a^5 + 61407232*a^6 + 7438336*a^7 + 524288*a
^8 + 16384*a^9 + 1358954496))/(256*(48384*a + 49248*a^2 + 28560*a^3 + 10321*a^4 + 2380*a^5 + 342*a^6 + 28*a^7
+ a^8 + 20736)))*((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/2) - 49*a^2*((a + 4)^15)^(1/
2) + 41598976*a^2 + 25672960*a^3 + 10187840*a^4 + 2695744*a^5 + 475608*a^6 + 53949*a^7 + 3570*a^8 + 105*a^9 +
16531456))/(16384*(1061683200*a + 2061434880*a^2 + 2474311680*a^3 + 2053201920*a^4 + 1247703040*a^5 + 57362176
0*a^6 + 203166720*a^7 + 55893360*a^8 + 11944200*a^9 + 1966491*a^10 + 244965*a^11 + 22350*a^12 + 1410*a^13 + 55
*a^14 + a^15 + 254803968)))^(1/2))*((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/2) - 49*a^
2*((a + 4)^15)^(1/2) + 41598976*a^2 + 25672960*a^3 + 10187840*a^4 + 2695744*a^5 + 475608*a^6 + 53949*a^7 + 357
0*a^8 + 105*a^9 + 16531456))/(16384*(1061683200*a + 2061434880*a^2 + 2474311680*a^3 + 2053201920*a^4 + 1247703
040*a^5 + 573621760*a^6 + 203166720*a^7 + 55893360*a^8 + 11944200*a^9 + 1966491*a^10 + 244965*a^11 + 22350*a^1
2 + 1410*a^13 + 55*a^14 + a^15 + 254803968)))^(1/2) + (108343296*a + 74059776*a^2 + 27065088*a^3 + 5576256*a^4
+ 614016*a^5 + 28224*a^6 + 66207744)/(16384*(940032*a + 1195776*a^2 + 899328*a^3 + 442864*a^4 + 149208*a^5 +
34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) - (x*(73476*a + 31545*a^2 + 6066*a^3 + 441*a^4 + 646
56))/(256*(48384*a + 49248*a^2 + 28560*a^3 + 10321*a^4 + 2380*a^5 + 342*a^6 + 28*a^7 + a^8 + 20736)))*((9*(393
29792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/2) - 49*a^2*((a + 4)^15)^(1/2) + 41598976*a^2 + 25672
960*a^3 + 10187840*a^4 + 2695744*a^5 + 475608*a^6 + 53949*a^7 + 3570*a^8 + 105*a^9 + 16531456))/(16384*(106168
3200*a + 2061434880*a^2 + 2474311680*a^3 + 2053201920*a^4 + 1247703040*a^5 + 573621760*a^6 + 203166720*a^7 + 5
5893360*a^8 + 11944200*a^9 + 1966491*a^10 + 244965*a^11 + 22350*a^12 + 1410*a^13 + 55*a^14 + a^15 + 254803968)
))^(1/2)*1i - (((52357496832*a + 57139003392*a^2 + 36322148352*a^3 + 14822473728*a^4 + 4027170816*a^5 + 728506
368*a^6 + 84615168*a^7 + 5726208*a^8 + 172032*a^9 + 21290287104)/(16384*(940032*a + 1195776*a^2 + 899328*a^3 +
442864*a^4 + 149208*a^5 + 34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) - ((4290672328704*a + 600
1143054336*a^2 + 5025917042688*a^3 + 2800520003584*a^4 + 1090200272896*a^5 + 302556119040*a^6 + 59862155264*a^
7 + 8275361792*a^8 + 761266176*a^9 + 41943040*a^10 + 1048576*a^11 + 1391569403904)/(16384*(940032*a + 1195776*
a^2 + 899328*a^3 + 442864*a^4 + 149208*a^5 + 34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) - (x*(3
510632448*a + 4020240384*a^2 + 2678587392*a^3 + 1144324096*a^4 + 325074944*a^5 + 61407232*a^6 + 7438336*a^7 +
524288*a^8 + 16384*a^9 + 1358954496))/(256*(48384*a + 49248*a^2 + 28560*a^3 + 10321*a^4 + 2380*a^5 + 342*a^6 +
28*a^7 + a^8 + 20736)))*((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/2) - 49*a^2*((a + 4)
^15)^(1/2) + 41598976*a^2 + 25672960*a^3 + 10187840*a^4 + 2695744*a^5 + 475608*a^6 + 53949*a^7 + 3570*a^8 + 10
5*a^9 + 16531456))/(16384*(1061683200*a + 2061434880*a^2 + 2474311680*a^3 + 2053201920*a^4 + 1247703040*a^5 +
573621760*a^6 + 203166720*a^7 + 55893360*a^8 + 11944200*a^9 + 1966491*a^10 + 244965*a^11 + 22350*a^12 + 1410*a
^13 + 55*a^14 + a^15 + 254803968)))^(1/2))*((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/2)
- 49*a^2*((a + 4)^15)^(1/2) + 41598976*a^2 + 25672960*a^3 + 10187840*a^4 + 2695744*a^5 + 475608*a^6 + 53949*a
^7 + 3570*a^8 + 105*a^9 + 16531456))/(16384*(1061683200*a + 2061434880*a^2 + 2474311680*a^3 + 2053201920*a^4 +
1247703040*a^5 + 573621760*a^6 + 203166720*a^7 + 55893360*a^8 + 11944200*a^9 + 1966491*a^10 + 244965*a^11 + 2
2350*a^12 + 1410*a^13 + 55*a^14 + a^15 + 254803968)))^(1/2) - (108343296*a + 74059776*a^2 + 27065088*a^3 + 557
6256*a^4 + 614016*a^5 + 28224*a^6 + 66207744)/(16384*(940032*a + 1195776*a^2 + 899328*a^3 + 442864*a^4 + 14920
8*a^5 + 34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) + (x*(73476*a + 31545*a^2 + 6066*a^3 + 441*a
^4 + 64656))/(256*(48384*a + 49248*a^2 + 28560*a^3 + 10321*a^4 + 2380*a^5 + 342*a^6 + 28*a^7 + a^8 + 20736)))*
((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/2) - 49*a^2*((a + 4)^15)^(1/2) + 41598976*a^2
+ 25672960*a^3 + 10187840*a^4 + 2695744*a^5 + ...
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