3.12.23 \(\int \frac {\sqrt {1-4 x+x^2}+(1-4 x+x^2)^{3/2}}{\sqrt {1-4 x+x^2}+(1-4 x+x^2)^{3/2}-(1-4 x+x^2)^{5/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4-8 \text {$\#$1}^3+17 \text {$\#$1}^2-4 \text {$\#$1}-1\& ,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})-4 \text {$\#$1} \log (x-\text {$\#$1})+2 \log (x-\text {$\#$1})}{2 \text {$\#$1}^3-12 \text {$\#$1}^2+17 \text {$\#$1}-2}\& \right ] \]

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Rubi [A]  time = 0.29, antiderivative size = 80, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6688, 1680, 1166, 207} \begin {gather*} \sqrt {\frac {1}{110} \left (17+7 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{7+\sqrt {5}}} (x-2)\right )-\sqrt {\frac {1}{110} \left (17-7 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{7-\sqrt {5}}} (x-2)\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2))/(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2) - (1 - 4*x + x^
2)^(5/2)),x]

[Out]

-(Sqrt[(17 - 7*Sqrt[5])/110]*ArcTanh[Sqrt[2/(7 - Sqrt[5])]*(-2 + x)]) + Sqrt[(17 + 7*Sqrt[5])/110]*ArcTanh[Sqr
t[2/(7 + Sqrt[5])]*(-2 + x)]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(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[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx &=\int \frac {2-4 x+x^2}{1+4 x-17 x^2+8 x^3-x^4} \, dx\\ &=\operatorname {Subst}\left (\int \frac {2-x^2}{11-7 x^2+x^4} \, dx,x,-2+x\right )\\ &=\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {7}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,-2+x\right )-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {7}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,-2+x\right )\\ &=\sqrt {\frac {1}{110} \left (17-7 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{7-\sqrt {5}}} (2-x)\right )-\sqrt {\frac {1}{110} \left (17+7 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{7+\sqrt {5}}} (2-x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4-8 \text {$\#$1}^3+17 \text {$\#$1}^2-4 \text {$\#$1}-1\&,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})-4 \text {$\#$1} \log (x-\text {$\#$1})+2 \log (x-\text {$\#$1})}{2 \text {$\#$1}^3-12 \text {$\#$1}^2+17 \text {$\#$1}-2}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2))/(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2) - (1 - 4*
x + x^2)^(5/2)),x]

[Out]

-1/2*RootSum[-1 - 4*#1 + 17*#1^2 - 8*#1^3 + #1^4 & , (2*Log[x - #1] - 4*Log[x - #1]*#1 + Log[x - #1]*#1^2)/(-2
+ 17*#1 - 12*#1^2 + 2*#1^3) & ]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2))/(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2)
- (1 - 4*x + x^2)^(5/2)),x]

[Out]

Could not integrate

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fricas [B]  time = 0.57, size = 163, normalized size = 1.96 \begin {gather*} \frac {1}{220} \, \sqrt {110} \sqrt {7 \, \sqrt {5} + 17} \log \left (\sqrt {110} \sqrt {7 \, \sqrt {5} + 17} {\left (4 \, \sqrt {5} - 5\right )} + 110 \, x - 220\right ) - \frac {1}{220} \, \sqrt {110} \sqrt {7 \, \sqrt {5} + 17} \log \left (-\sqrt {110} \sqrt {7 \, \sqrt {5} + 17} {\left (4 \, \sqrt {5} - 5\right )} + 110 \, x - 220\right ) - \frac {1}{220} \, \sqrt {110} \sqrt {-7 \, \sqrt {5} + 17} \log \left (\sqrt {110} {\left (4 \, \sqrt {5} + 5\right )} \sqrt {-7 \, \sqrt {5} + 17} + 110 \, x - 220\right ) + \frac {1}{220} \, \sqrt {110} \sqrt {-7 \, \sqrt {5} + 17} \log \left (-\sqrt {110} {\left (4 \, \sqrt {5} + 5\right )} \sqrt {-7 \, \sqrt {5} + 17} + 110 \, x - 220\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x, alg
orithm="fricas")

[Out]

1/220*sqrt(110)*sqrt(7*sqrt(5) + 17)*log(sqrt(110)*sqrt(7*sqrt(5) + 17)*(4*sqrt(5) - 5) + 110*x - 220) - 1/220
*sqrt(110)*sqrt(7*sqrt(5) + 17)*log(-sqrt(110)*sqrt(7*sqrt(5) + 17)*(4*sqrt(5) - 5) + 110*x - 220) - 1/220*sqr
t(110)*sqrt(-7*sqrt(5) + 17)*log(sqrt(110)*(4*sqrt(5) + 5)*sqrt(-7*sqrt(5) + 17) + 110*x - 220) + 1/220*sqrt(1
10)*sqrt(-7*sqrt(5) + 17)*log(-sqrt(110)*(4*sqrt(5) + 5)*sqrt(-7*sqrt(5) + 17) + 110*x - 220)

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giac [B]  time = 0.40, size = 349, normalized size = 4.20 \begin {gather*} \frac {{\left ({\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{3} + 12 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 17 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 32\right )}} - \frac {{\left ({\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} - 4 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{3} - 12 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} + 17 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 32\right )}} + \frac {{\left ({\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 4 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x + \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{3} + 12 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 17 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 32\right )}} - \frac {{\left ({\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} - 4 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x - \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{3} - 12 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} + 17 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x, alg
orithm="giac")

[Out]

1/2*((sqrt(1/2*sqrt(5) + 7/2) - 2)^2 + 4*sqrt(1/2*sqrt(5) + 7/2) - 6)*log(x + sqrt(1/2*sqrt(5) + 7/2) - 2)/(2*
(sqrt(1/2*sqrt(5) + 7/2) - 2)^3 + 12*(sqrt(1/2*sqrt(5) + 7/2) - 2)^2 + 17*sqrt(1/2*sqrt(5) + 7/2) - 32) - 1/2*
((sqrt(1/2*sqrt(5) + 7/2) + 2)^2 - 4*sqrt(1/2*sqrt(5) + 7/2) - 6)*log(x - sqrt(1/2*sqrt(5) + 7/2) - 2)/(2*(sqr
t(1/2*sqrt(5) + 7/2) + 2)^3 - 12*(sqrt(1/2*sqrt(5) + 7/2) + 2)^2 + 17*sqrt(1/2*sqrt(5) + 7/2) + 32) + 1/2*((sq
rt(-1/2*sqrt(5) + 7/2) - 2)^2 + 4*sqrt(-1/2*sqrt(5) + 7/2) - 6)*log(x + sqrt(-1/2*sqrt(5) + 7/2) - 2)/(2*(sqrt
(-1/2*sqrt(5) + 7/2) - 2)^3 + 12*(sqrt(-1/2*sqrt(5) + 7/2) - 2)^2 + 17*sqrt(-1/2*sqrt(5) + 7/2) - 32) - 1/2*((
sqrt(-1/2*sqrt(5) + 7/2) + 2)^2 - 4*sqrt(-1/2*sqrt(5) + 7/2) - 6)*log(x - sqrt(-1/2*sqrt(5) + 7/2) - 2)/(2*(sq
rt(-1/2*sqrt(5) + 7/2) + 2)^3 - 12*(sqrt(-1/2*sqrt(5) + 7/2) + 2)^2 + 17*sqrt(-1/2*sqrt(5) + 7/2) + 32)

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maple [B]  time = 1.09, size = 57, normalized size = 0.69

method	result	size
risch	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-8 \textit {\_Z}^{3}+17 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-4 \textit {\_R} +2\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-12 \textit {\_R}^{2}+17 \textit {\_R} -2}\right )}{2}\)	\(57\)
trager	\(-\RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {440 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{3} x +440 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x -122 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right ) x -10 x +7}{440 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{3} x -440 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x -122 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right ) x +10 x -7}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+12100 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) \ln \left (-\frac {20 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+12100 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x +2200 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x +4 \RootOf \left (\textit {\_Z}^{2}+12100 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x -120 x -35}{20 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+12100 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x -2200 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2} x +4 \RootOf \left (\textit {\_Z}^{2}+12100 \RootOf \left (4400 \textit {\_Z}^{4}-340 \textit {\_Z}^{2}+1\right )^{2}-935\right ) x +120 x +35}\right )}{110}\)	\(345\)
default	error in AlgebraicFunction: argument is not an algebraic\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x,method=_RE
TURNVERBOSE)

[Out]

-1/2*sum((_R^2-4*_R+2)/(2*_R^3-12*_R^2+17*_R-2)*ln(x-_R),_R=RootOf(_Z^4-8*_Z^3+17*_Z^2-4*_Z-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{2} - 4 \, x + 1\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 4 \, x + 1}}{{\left (x^{2} - 4 \, x + 1\right )}^{\frac {5}{2}} - {\left (x^{2} - 4 \, x + 1\right )}^{\frac {3}{2}} - \sqrt {x^{2} - 4 \, x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2))/((x^2-4*x+1)^(1/2)+(x^2-4*x+1)^(3/2)-(x^2-4*x+1)^(5/2)),x, alg
orithm="maxima")

[Out]

-integrate(((x^2 - 4*x + 1)^(3/2) + sqrt(x^2 - 4*x + 1))/((x^2 - 4*x + 1)^(5/2) - (x^2 - 4*x + 1)^(3/2) - sqrt
(x^2 - 4*x + 1)), x)

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mupad [B]  time = 0.92, size = 425, normalized size = 5.12 \begin {gather*} \frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )\,\left (2\,\sqrt {2}\,\sqrt {7-\sqrt {5}}+{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )}^2-6\right )}{17\,\sqrt {2}\,\sqrt {7-\sqrt {5}}+24\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )}^2+4\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )}^3-64}+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}+7}-{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}+2\right )}^2+6\right )}{4\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}+2\right )}^3-24\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}+2\right )}^2+17\,\sqrt {2}\,\sqrt {\sqrt {5}+7}+64}+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}-2\right )\,\left (2\,\sqrt {2}\,\sqrt {7-\sqrt {5}}-{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}+2\right )}^2+6\right )}{17\,\sqrt {2}\,\sqrt {7-\sqrt {5}}-24\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}+2\right )}^2+4\,{\left (\frac {\sqrt {2}\,\sqrt {7-\sqrt {5}}}{2}+2\right )}^3+64}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )\,\left ({\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )}^2+2\,\sqrt {2}\,\sqrt {\sqrt {5}+7}-6\right )}{24\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )}^2+4\,{\left (\frac {\sqrt {2}\,\sqrt {\sqrt {5}+7}}{2}-2\right )}^3+17\,\sqrt {2}\,\sqrt {\sqrt {5}+7}-64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 4*x + 1)^(1/2) + (x^2 - 4*x + 1)^(3/2))/((x^2 - 4*x + 1)^(1/2) + (x^2 - 4*x + 1)^(3/2) - (x^2 - 4*
x + 1)^(5/2)),x)

[Out]

(log(x + (2^(1/2)*(7 - 5^(1/2))^(1/2))/2 - 2)*(2*2^(1/2)*(7 - 5^(1/2))^(1/2) + ((2^(1/2)*(7 - 5^(1/2))^(1/2))/
2 - 2)^2 - 6))/(17*2^(1/2)*(7 - 5^(1/2))^(1/2) + 24*((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 - 2)^2 + 4*((2^(1/2)*(7 -
5^(1/2))^(1/2))/2 - 2)^3 - 64) + (log(x - (2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)*(2*2^(1/2)*(5^(1/2) + 7)^(1/2)
- ((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 + 2)^2 + 6))/(4*((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 + 2)^3 - 24*((2^(1/2)*(5^
(1/2) + 7)^(1/2))/2 + 2)^2 + 17*2^(1/2)*(5^(1/2) + 7)^(1/2) + 64) + (log(x - (2^(1/2)*(7 - 5^(1/2))^(1/2))/2 -
2)*(2*2^(1/2)*(7 - 5^(1/2))^(1/2) - ((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 + 2)^2 + 6))/(17*2^(1/2)*(7 - 5^(1/2))^(
1/2) - 24*((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 + 2)^2 + 4*((2^(1/2)*(7 - 5^(1/2))^(1/2))/2 + 2)^3 + 64) + (log(x +
(2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)*(((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)^2 + 2*2^(1/2)*(5^(1/2) + 7)^(1/2)
- 6))/(24*((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)^2 + 4*((2^(1/2)*(5^(1/2) + 7)^(1/2))/2 - 2)^3 + 17*2^(1/2)*(5
^(1/2) + 7)^(1/2) - 64)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SympifyError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-4*x+1)**(1/2)+(x**2-4*x+1)**(3/2))/((x**2-4*x+1)**(1/2)+(x**2-4*x+1)**(3/2)-(x**2-4*x+1)**(5/
2)),x)

[Out]

Exception raised: SympifyError

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