3.122 \(\int \frac{1}{(a+8 x-8 x^2+4 x^3-x^4)^3} \, dx\)
Optimal. Leaf size=252 \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}-\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )} \]
[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]])
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Rubi [A] time = 0.531924, antiderivative size = 252, normalized size of antiderivative = 1.,
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 =
{1106, 1092, 1178, 1166, 204} \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}-\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )} \]
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 1106
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 1092
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 1178
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]
Rule 1166
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 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])
Rubi steps
\begin{align*} \int \frac{1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx &=\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.118802, size = 254, normalized size = 1.01 \[ \frac{1}{128} \left (-\frac{3 \text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\& ,\frac{4 \text{$\#$1}^2 a \log (x-\text{$\#$1})+14 \text{$\#$1}^2 \log (x-\text{$\#$1})+7 a^2 \log (x-\text{$\#$1})+55 a \log (x-\text{$\#$1})-8 \text{$\#$1} a \log (x-\text{$\#$1})+108 \log (x-\text{$\#$1})-28 \text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\& \right ]}{\left (a^2+7 a+12\right )^2}+\frac{4 (x-1) \left (7 a^2+a \left (12 x^2-24 x+79\right )+6 \left (7 x^2-14 x+32\right )\right )}{(a+3)^2 (a+4)^2 \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}+\frac{16 (x-1) \left (a+x^2-2 x+6\right )}{(a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )^2}\right ) \]
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] time = 0.016, size = 398, normalized size = 1.6 \begin{align*} -{\frac{1}{ \left ({x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x \right ) ^{2}} \left ({\frac{ \left ( 6\,a+21 \right ){x}^{7}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 147+42\,a \right ){x}^{6}}{ \left ( 16\,{a}^{2}+128\,a+256 \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{ \left ( 35\,{a}^{2}+875\,a+2640 \right ){x}^{4}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}+{\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\,{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\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}} \right ) }-{\frac{3}{ \left ( 128\,{a}^{3}+1280\,{a}^{2}+4224\,a+4608 \right ) \left ( 4+a \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( 108+2\, \left ( 7+2\,a \right ){{\it \_R}}^{2}+4\, \left ( -2\,a-7 \right ){\it \_R}+7\,{a}^{2}+55\,a \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2}}} \end{align*}
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)
[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)/(a^4+14*a^3+73*a^2+
168*a+144))/(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*(7+2*a)*_R^2+4*(-2*a-7)*_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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
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]
Exception raised: AttributeError
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Fricas [B] time = 1.90238, size = 13437, normalized size = 53.32 \begin{align*} \text{result too large to display} \end{align*}
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^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9
216)*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 +
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 + 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 + 161
44)/(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 + 7257
60*a + 248832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^3 + 177061*a^2 + 415884*a +
367536)*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^1
1 + 8881*a^10 + 103320*a^9 + 810205*a^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*a^4 + 18123840
0*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 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176))
+ 6613472*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 +
129367*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 + 2
8570320*a^8 + 94320045*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 2
77136640*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 3730
20*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 9923472) + 3*((a^4 + 14*a^3 + 73*a^2 + 1
68*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
+ 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 + 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 + 28570320*a^8 + 94320045*a^7 + 2418700
50*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 + 7
25760*a + 248832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^3 + 177061*a^2 + 415884*
a + 367536)*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 + 18123
8400*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 + 9432004
5*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176
)) + 6613472*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 + 129367*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
+ 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 + 3
73020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 9923472) + 524*a^2 - 4*(11*a^3 + 107*
a^2 - 84*a - 1152)*x + 1632*a + 1152)/((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 + 14
4)*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)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
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: TypeError