3.120 \(\int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx\)
Optimal. Leaf size=89 \[ \frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {a+4} \sqrt {\sqrt {a+4}+1}}-\frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {a+4} \sqrt {1-\sqrt {a+4}}} \]
[Out]
-1/2*arctan((-1+x)/(1-(4+a)^(1/2))^(1/2))/(4+a)^(1/2)/(1-(4+a)^(1/2))^(1/2)+1/2*arctan((-1+x)/(1+(4+a)^(1/2))^
(1/2))/(4+a)^(1/2)/(1+(4+a)^(1/2))^(1/2)
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used =
{1106, 1093, 204} \[ \frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {a+4} \sqrt {\sqrt {a+4}+1}}-\frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {a+4} \sqrt {1-\sqrt {a+4}}} \]
Antiderivative was successfully verified.
[In]
Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-1),x]
[Out]
-ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 - Sqrt[4 + a]]) + ArcTan[(-1 + x)/Sqrt[1 + Sqrt[
4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 + Sqrt[4 + a]])
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])
Rule 1093
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]
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]
Rubi steps
\begin {align*} \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx &=\operatorname {Subst}\left (\int \frac {1}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{2 \sqrt {4+a}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{2 \sqrt {4+a}}\\ &=\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{2 \sqrt {4+a} \sqrt {1-\sqrt {4+a}}}-\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{2 \sqrt {4+a} \sqrt {1+\sqrt {4+a}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.64 \[ -\frac {1}{4} \text {RootSum}\left [-\text {$\#$1}^4+4 \text {$\#$1}^3-8 \text {$\#$1}^2+8 \text {$\#$1}+a\& ,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}^3-3 \text {$\#$1}^2+4 \text {$\#$1}-2}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-1),x]
[Out]
-1/4*RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , Log[x - #1]/(-2 + 4*#1 - 3*#1^2 + #1^3) & ]
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fricas [B] time = 0.45, size = 457, normalized size = 5.13 \[ \frac {1}{4} \, \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} \log \left ({\left (a - \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) - \frac {1}{4} \, \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} \log \left (-{\left (a - \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) + \frac {1}{4} \, \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} \log \left ({\left (a + \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) - \frac {1}{4} \, \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} \log \left (-{\left (a + \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x^4+4*x^3-8*x^2+a+8*x),x, algorithm="fricas")
[Out]
1/4*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12))*log((a - (a^2 + 7*a + 12)/sqr
t(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12))
+ x - 1) - 1/4*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12))*log(-(a - (a^2 + 7
*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 +
7*a + 12)) + x - 1) + 1/4*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12))*log((
a + (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36
) - 1)/(a^2 + 7*a + 12)) + x - 1) - 1/4*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) - 1)/(a^2 + 7*a
+ 12))*log(-(a + (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a
^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12)) + x - 1)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x^4+4*x^3-8*x^2+a+8*x),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 [a]=[86]Warning, need
to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assum
ing [a]=[12]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong
.The choice was done assuming [a]=[48]Warning, need to choose a branch for the root of a polynomial with param
eters. This might be wrong.The choice was done assuming [a]=[34]-sqrt(1/16)*sqrt(1/256*(256*sqrt(a+4)*(a+4)-25
6*a-1024)/(-a^3-11*a^2-40*a-48))*ln(abs(-a^5*sqrt(a+4)+a^5+a^4*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x-a^4*
sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-17*a^4*sqrt(a+4)+17*a^4-a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*
sqrt(a+4)*x+a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)+14*a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a
+12)*x-14*a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-111*a^3*sqrt(a+4)+111*a^3-10*a^2*sqrt(sqrt(a+4)*(-a^2-7
*a-12)+a^2+7*a+12)*sqrt(a+4)*x+10*a^2*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)+69*a^2*sqrt(sqrt(a+4)
*(-a^2-7*a-12)+a^2+7*a+12)*x-69*a^2*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-351*a^2*sqrt(a+4)+351*a^2-29*a*sq
rt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)*x+29*a*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)+144
*a*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x-144*a*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-544*a*sqrt(a+4)+5
44*a-28*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)*x+28*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(
a+4)+112*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x-112*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-336*sqrt(a+4)
+336))+sqrt(1/16)*sqrt(1/256*(256*sqrt(a+4)*(a+4)-256*a-1024)/(-a^3-11*a^2-40*a-48))*ln(abs(-a^5*sqrt(a+4)+a^5
-a^4*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x+a^4*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-17*a^4*sqrt(a+4)+
17*a^4+a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)*x-a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*s
qrt(a+4)-14*a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x+14*a^3*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-111
*a^3*sqrt(a+4)+111*a^3+10*a^2*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)*x-10*a^2*sqrt(sqrt(a+4)*(-a^2
-7*a-12)+a^2+7*a+12)*sqrt(a+4)-69*a^2*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x+69*a^2*sqrt(sqrt(a+4)*(-a^2-7
*a-12)+a^2+7*a+12)-351*a^2*sqrt(a+4)+351*a^2+29*a*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)*x-29*a*sq
rt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)-144*a*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x+144*a*sqrt(s
qrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-544*a*sqrt(a+4)+544*a+28*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)
*x-28*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*sqrt(a+4)-112*sqrt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)*x+112*sq
rt(sqrt(a+4)*(-a^2-7*a-12)+a^2+7*a+12)-336*sqrt(a+4)+336))-sqrt(1/16)*sqrt(1/256*(-256*sqrt(a+4)*(a+4)-256*a-1
024)/(-a^3-11*a^2-40*a-48))*ln(abs(a^5*sqrt(a+4)+a^5+a^4*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*x-a^4*sqrt(sq
rt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+17*a^4*sqrt(a+4)+17*a^4+a^3*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)
*x-a^3*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)+14*a^3*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*x-14*a
^3*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+111*a^3*sqrt(a+4)+111*a^3+10*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*
a+12)*sqrt(a+4)*x-10*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)+69*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+
a^2+7*a+12)*x-69*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+351*a^2*sqrt(a+4)+351*a^2+29*a*sqrt(sqrt(a+4)*(a^
2+7*a+12)+a^2+7*a+12)*sqrt(a+4)*x-29*a*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)+144*a*sqrt(sqrt(a+4)*
(a^2+7*a+12)+a^2+7*a+12)*x-144*a*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+544*a*sqrt(a+4)+544*a+28*sqrt(sqrt(a+
4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)*x-28*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)+112*sqrt(sqrt(a+4
)*(a^2+7*a+12)+a^2+7*a+12)*x-112*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+336*sqrt(a+4)+336))+sqrt(1/16)*sqrt(1
/256*(-256*sqrt(a+4)*(a+4)-256*a-1024)/(-a^3-11*a^2-40*a-48))*ln(abs(a^5*sqrt(a+4)+a^5-a^4*sqrt(sqrt(a+4)*(a^2
+7*a+12)+a^2+7*a+12)*x+a^4*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+17*a^4*sqrt(a+4)+17*a^4-a^3*sqrt(sqrt(a+4)*
(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)*x+a^3*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)-14*a^3*sqrt(sqrt(a+
4)*(a^2+7*a+12)+a^2+7*a+12)*x+14*a^3*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+111*a^3*sqrt(a+4)+111*a^3-10*a^2*
sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)*x+10*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)-6
9*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*x+69*a^2*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+351*a^2*sqrt(a+
4)+351*a^2-29*a*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)*x+29*a*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+1
2)*sqrt(a+4)-144*a*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*x+144*a*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+544
*a*sqrt(a+4)+544*a-28*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*sqrt(a+4)*x+28*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7
*a+12)*sqrt(a+4)-112*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)*x+112*sqrt(sqrt(a+4)*(a^2+7*a+12)+a^2+7*a+12)+336
*sqrt(a+4)+336))
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maple [C] time = 0.02, size = 49, normalized size = 0.55 \[ -\frac {\ln \left (-\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )+x \right )}{4 \left (\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{3}-3 \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{2}+4 \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-x^4+4*x^3-8*x^2+a+8*x),x)
[Out]
-1/4*sum(1/(_R^3-3*_R^2+4*_R-2)*ln(-_R+x),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x^4+4*x^3-8*x^2+a+8*x),x, algorithm="maxima")
[Out]
-integrate(1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)
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mupad [B] time = 2.58, size = 571, normalized size = 6.42 \[ -\mathrm {atan}\left (-\frac {a\,8{}\mathrm {i}-x\,16{}\mathrm {i}+x\,\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a\,x\,8{}\mathrm {i}-\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a^2\,x\,1{}\mathrm {i}+a^2\,1{}\mathrm {i}+16{}\mathrm {i}}{44\,a^2\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+4\,a^3\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+160\,a\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+192\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}}\right )\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}\,2{}\mathrm {i}-\mathrm {atan}\left (-\frac {a\,8{}\mathrm {i}-x\,16{}\mathrm {i}-x\,\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a\,x\,8{}\mathrm {i}+\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a^2\,x\,1{}\mathrm {i}+a^2\,1{}\mathrm {i}+16{}\mathrm {i}}{160\,a\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+192\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+44\,a^2\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+4\,a^3\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}}\right )\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + 8*x - 8*x^2 + 4*x^3 - x^4),x)
[Out]
- atan(-(a*8i - x*16i + x*(48*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a*x*8i - (48*a + 12*a^2 + a^3 + 64)^(1/2)*1i -
a^2*x*1i + a^2*1i + 16i)/(44*a^2*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768)
)^(1/2) + 4*a^3*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2) + 160*a*((
a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2) + 192*((a - (48*a + 12*a^2 +
a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)))*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(6
40*a + 176*a^2 + 16*a^3 + 768))^(1/2)*2i - atan(-(a*8i - x*16i - x*(48*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a*x*8
i + (48*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a^2*x*1i + a^2*1i + 16i)/(160*a*((a + (48*a + 12*a^2 + a^3 + 64)^(1/
2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2) + 192*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176
*a^2 + 16*a^3 + 768))^(1/2) + 44*a^2*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 7
68))^(1/2) + 4*a^3*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)))*((a +
(48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)*2i
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sympy [A] time = 0.93, size = 66, normalized size = 0.74 \[ - \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} + 2816 a^{2} + 10240 a + 12288\right ) + t^{2} \left (- 32 a - 128\right ) - 1, \left (t \mapsto t \log {\left (64 t^{3} a^{2} + 448 t^{3} a + 768 t^{3} - 4 t a - 20 t + x - 1 \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-x**4+4*x**3-8*x**2+a+8*x),x)
[Out]
-RootSum(_t**4*(256*a**3 + 2816*a**2 + 10240*a + 12288) + _t**2*(-32*a - 128) - 1, Lambda(_t, _t*log(64*_t**3*
a**2 + 448*_t**3*a + 768*_t**3 - 4*_t*a - 20*_t + x - 1)))
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