3.390 \(\int \frac{x^2}{2+(1-x^2)^4} \, dx\)
Optimal. Leaf size=188 \[ -\frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{1}{8} i \left (\sqrt [4]{-2}+\sqrt{2}\right ) \sqrt{\frac{1+i}{2^{3/4}+(1+i)}} \tanh ^{-1}\left (\sqrt{\frac{1+i}{2^{3/4}+(1+i)}} x\right ) \]
[Out]
-((-1)^(1/4)*Sqrt[1 - (-2)^(1/4)]*ArcTanh[x/Sqrt[1 - (-2)^(1/4)]])/(4*2^(3/4)) + ((-1)^(3/4)*Sqrt[1 + I*(-2)^(
1/4)]*ArcTanh[x/Sqrt[1 + I*(-2)^(1/4)]])/(4*2^(3/4)) + ((-1)^(1/4)*Sqrt[1 + (-2)^(1/4)]*ArcTanh[x/Sqrt[1 + (-2
)^(1/4)]])/(4*2^(3/4)) - (I/8)*((-2)^(1/4) + Sqrt[2])*Sqrt[(1 + I)/((1 + I) + 2^(3/4))]*ArcTanh[Sqrt[(1 + I)/(
(1 + I) + 2^(3/4))]*x]
________________________________________________________________________________________
Rubi [A] time = 0.155724, antiderivative size = 188, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used =
{6740, 206, 207, 1972, 208} \[ -\frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{1}{8} i \left (\sqrt [4]{-2}+\sqrt{2}\right ) \sqrt{\frac{1+i}{2^{3/4}+(1+i)}} \tanh ^{-1}\left (\sqrt{\frac{1+i}{2^{3/4}+(1+i)}} x\right ) \]
Antiderivative was successfully verified.
[In]
Int[x^2/(2 + (1 - x^2)^4),x]
[Out]
-((-1)^(1/4)*Sqrt[1 - (-2)^(1/4)]*ArcTanh[x/Sqrt[1 - (-2)^(1/4)]])/(4*2^(3/4)) + ((-1)^(3/4)*Sqrt[1 + I*(-2)^(
1/4)]*ArcTanh[x/Sqrt[1 + I*(-2)^(1/4)]])/(4*2^(3/4)) + ((-1)^(1/4)*Sqrt[1 + (-2)^(1/4)]*ArcTanh[x/Sqrt[1 + (-2
)^(1/4)]])/(4*2^(3/4)) - (I/8)*((-2)^(1/4) + Sqrt[2])*Sqrt[(1 + I)/((1 + I) + 2^(3/4))]*ArcTanh[Sqrt[(1 + I)/(
(1 + I) + 2^(3/4))]*x]
Rule 6740
Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
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 1972
Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] && !BinomialMatchQ[
u, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{x^2}{2+\left (1-x^2\right )^4} \, dx &=\int \left (\frac{\sqrt [4]{-2}+i \sqrt{2}}{8 \left (1+\sqrt [4]{-2}-x^2\right )}+\frac{\sqrt [4]{-2}-i \sqrt{2}}{8 \left (-1+\sqrt [4]{-2}+x^2\right )}+\frac{\sqrt [4]{-2}-\sqrt{2}}{8 \left (\sqrt [4]{-2}-i \left (1-x^2\right )\right )}+\frac{\sqrt [4]{-2}+\sqrt{2}}{8 \left (\sqrt [4]{-2}+i \left (1-x^2\right )\right )}\right ) \, dx\\ &=\frac{1}{8} \left (\sqrt [4]{-2}-\sqrt{2}\right ) \int \frac{1}{\sqrt [4]{-2}-i \left (1-x^2\right )} \, dx+\frac{1}{8} \left (\sqrt [4]{-2}-i \sqrt{2}\right ) \int \frac{1}{-1+\sqrt [4]{-2}+x^2} \, dx+\frac{1}{8} \left (\sqrt [4]{-2}+i \sqrt{2}\right ) \int \frac{1}{1+\sqrt [4]{-2}-x^2} \, dx+\frac{1}{8} \left (\sqrt [4]{-2}+\sqrt{2}\right ) \int \frac{1}{\sqrt [4]{-2}+i \left (1-x^2\right )} \, dx\\ &=-\frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{1}{8} \left (\sqrt [4]{-2}-\sqrt{2}\right ) \int \frac{1}{-i+\sqrt [4]{-2}+i x^2} \, dx+\frac{1}{8} \left (\sqrt [4]{-2}+\sqrt{2}\right ) \int \frac{1}{i+\sqrt [4]{-2}-i x^2} \, dx\\ &=-\frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{1}{8} i \left (\sqrt [4]{-2}+\sqrt{2}\right ) \sqrt{\frac{1+i}{(1+i)+2^{3/4}}} \tanh ^{-1}\left (\sqrt{\frac{1+i}{(1+i)+2^{3/4}}} x\right )\\ \end{align*}
Mathematica [C] time = 0.0132833, size = 61, normalized size = 0.32 \[ \frac{1}{8} \text{RootSum}\left [\text{$\#$1}^8-4 \text{$\#$1}^6+6 \text{$\#$1}^4-4 \text{$\#$1}^2+3\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^6-3 \text{$\#$1}^4+3 \text{$\#$1}^2-1}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(2 + (1 - x^2)^4),x]
[Out]
RootSum[3 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[x - #1]*#1)/(-1 + 3*#1^2 - 3*#1^4 + #1^6) & ]/8
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Maple [C] time = 0.006, size = 56, normalized size = 0.3 \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-4\,{{\it \_Z}}^{6}+6\,{{\it \_Z}}^{4}-4\,{{\it \_Z}}^{2}+3 \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}-3\,{{\it \_R}}^{5}+3\,{{\it \_R}}^{3}-{\it \_R}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(2+(-x^2+1)^4),x)
[Out]
1/8*sum(_R^2/(_R^7-3*_R^5+3*_R^3-_R)*ln(x-_R),_R=RootOf(_Z^8-4*_Z^6+6*_Z^4-4*_Z^2+3))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (x^{2} - 1\right )}^{4} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(2+(-x^2+1)^4),x, algorithm="maxima")
[Out]
integrate(x^2/((x^2 - 1)^4 + 2), x)
________________________________________________________________________________________
Fricas [B] time = 7.33124, size = 8735, normalized size = 46.46 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(2+(-x^2+1)^4),x, algorithm="fricas")
[Out]
-1/16*sqrt(2)*sqrt(sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1
/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*s
qrt(2))) - 1) + 32*sqrt(1/8192*I*sqrt(2)) + 32*sqrt(-1/8192*I*sqrt(2)))*log((16384*sqrt(2)*(-1/256*I*sqrt(2) -
1/2*sqrt(1/8192*I*sqrt(2)))^2*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) + 16384*(sqrt(2)*(I*sqrt(2) + 128*sq
rt(1/8192*I*sqrt(2))) + sqrt(2))*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 + 16384*sqrt(2)*(-1/256*I*s
qrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288
*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2)
+ 128*sqrt(-1/8192*I*sqrt(2))) - 1)*((sqrt(2)*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2))) + sqrt(2))*(-I*sqrt(2)
+ 128*sqrt(-1/8192*I*sqrt(2))) + sqrt(2)*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))) - sqrt(2))*sqrt(sqrt(-12288
*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 -
1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 1) + 32*sqrt(1/8192
*I*sqrt(2)) + 32*sqrt(-1/8192*I*sqrt(2))) + 2*x) + 1/16*sqrt(2)*sqrt(sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-
1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8
192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 1) + 32*sqrt(1/8192*I*sqrt(2)) + 32*sqrt(-1/8192*
I*sqrt(2)))*log(-(16384*sqrt(2)*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2*(-I*sqrt(2) + 128*sqrt(-1/81
92*I*sqrt(2))) + 16384*(sqrt(2)*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2))) + sqrt(2))*(1/256*I*sqrt(2) - 1/2*sqr
t(-1/8192*I*sqrt(2)))^2 + 16384*sqrt(2)*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - sqrt(-12288*(1/256
*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I
*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 1)*((sqrt(2)*(I*sqrt(2) +
128*sqrt(1/8192*I*sqrt(2))) + sqrt(2))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) + sqrt(2)*(I*sqrt(2) + 128*s
qrt(1/8192*I*sqrt(2)))) - sqrt(2))*sqrt(sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*
(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) +
128*sqrt(-1/8192*I*sqrt(2))) - 1) + 32*sqrt(1/8192*I*sqrt(2)) + 32*sqrt(-1/8192*I*sqrt(2))) + 2*x) - 1/16*sqr
t(2)*sqrt(-sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(
1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2)))
- 1) + 32*sqrt(1/8192*I*sqrt(2)) + 32*sqrt(-1/8192*I*sqrt(2)))*log((16384*sqrt(2)*(-1/256*I*sqrt(2) - 1/2*sqr
t(1/8192*I*sqrt(2)))^2*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) + 16384*(sqrt(2)*(I*sqrt(2) + 128*sqrt(1/819
2*I*sqrt(2))) + sqrt(2))*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 + 16384*sqrt(2)*(-1/256*I*sqrt(2) -
1/2*sqrt(1/8192*I*sqrt(2)))^2 + sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256
*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sq
rt(-1/8192*I*sqrt(2))) - 1)*((sqrt(2)*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2))) + sqrt(2))*(-I*sqrt(2) + 128*sq
rt(-1/8192*I*sqrt(2))) + sqrt(2)*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))) - sqrt(2))*sqrt(-sqrt(-12288*(1/256
*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I
*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 1) + 32*sqrt(1/8192*I*sqrt
(2)) + 32*sqrt(-1/8192*I*sqrt(2))) + 2*x) + 1/16*sqrt(2)*sqrt(-sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192
*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*
sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 1) + 32*sqrt(1/8192*I*sqrt(2)) + 32*sqrt(-1/8192*I*sqrt
(2)))*log(-(16384*sqrt(2)*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2*(-I*sqrt(2) + 128*sqrt(-1/8192*I*s
qrt(2))) + 16384*(sqrt(2)*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2))) + sqrt(2))*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8
192*I*sqrt(2)))^2 + 16384*sqrt(2)*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 + sqrt(-12288*(1/256*I*sqr
t(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(
2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 1)*((sqrt(2)*(I*sqrt(2) + 128*sq
rt(1/8192*I*sqrt(2))) + sqrt(2))*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) + sqrt(2)*(I*sqrt(2) + 128*sqrt(1/
8192*I*sqrt(2)))) - sqrt(2))*sqrt(-sqrt(-12288*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2 - 12288*(-1/2
56*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 1/8*(I*sqrt(2) + 128*sqrt(1/8192*I*sqrt(2)))*(-I*sqrt(2) + 128*
sqrt(-1/8192*I*sqrt(2))) - 1) + 32*sqrt(1/8192*I*sqrt(2)) + 32*sqrt(-1/8192*I*sqrt(2))) + 2*x) - sqrt(1/256*I*
sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))*log(8*(8388608*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^3 + 3276
8*(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2*(-I*sqrt(2) - 128*sqrt(1/8192*I*sqrt(2)) - 1) - 32768*(-1/
256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 2*I*sqrt(2) - 256*s
qrt(1/8192*I*sqrt(2)) - 3)*sqrt(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2))) + x) + sqrt(1/256*I*sqrt(2) - 1
/2*sqrt(-1/8192*I*sqrt(2)))*log(-8*(8388608*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^3 + 32768*(1/256*I
*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2)))^2*(-I*sqrt(2) - 128*sqrt(1/8192*I*sqrt(2)) - 1) - 32768*(-1/256*I*sqrt
(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2*(-I*sqrt(2) + 128*sqrt(-1/8192*I*sqrt(2))) - 2*I*sqrt(2) - 256*sqrt(1/8192
*I*sqrt(2)) - 3)*sqrt(1/256*I*sqrt(2) - 1/2*sqrt(-1/8192*I*sqrt(2))) + x) + sqrt(-1/256*I*sqrt(2) - 1/2*sqrt(1
/8192*I*sqrt(2)))*log(8*(8388608*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^3 + 32768*(-1/256*I*sqrt(2) -
1/2*sqrt(1/8192*I*sqrt(2)))^2 - 2*I*sqrt(2) - 256*sqrt(1/8192*I*sqrt(2)) - 5)*sqrt(-1/256*I*sqrt(2) - 1/2*sqr
t(1/8192*I*sqrt(2))) + x) - sqrt(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))*log(-8*(8388608*(-1/256*I*sqrt
(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^3 + 32768*(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2)))^2 - 2*I*sqrt(2) -
256*sqrt(1/8192*I*sqrt(2)) - 5)*sqrt(-1/256*I*sqrt(2) - 1/2*sqrt(1/8192*I*sqrt(2))) + x)
________________________________________________________________________________________
Sympy [A] time = 0.187626, size = 39, normalized size = 0.21 \begin{align*} \operatorname{RootSum}{\left (1073741824 t^{8} + 65536 t^{4} - 1024 t^{2} + 3, \left ( t \mapsto t \log{\left (67108864 t^{7} + 262144 t^{5} + 4096 t^{3} - 40 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(2+(-x**2+1)**4),x)
[Out]
RootSum(1073741824*_t**8 + 65536*_t**4 - 1024*_t**2 + 3, Lambda(_t, _t*log(67108864*_t**7 + 262144*_t**5 + 409
6*_t**3 - 40*_t + x)))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (x^{2} - 1\right )}^{4} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(2+(-x^2+1)^4),x, algorithm="giac")
[Out]
integrate(x^2/((x^2 - 1)^4 + 2), x)