3.389 \(\int \frac{x^2}{2+(1+x^2)^4} \, dx\)
Optimal. Leaf size=188 \[ \frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{1}{8} i \left (\sqrt [4]{-2}+\sqrt{2}\right ) \sqrt{\frac{1+i}{2^{3/4}+(1+i)}} \tan ^{-1}\left (\sqrt{\frac{1+i}{2^{3/4}+(1+i)}} x\right ) \]
[Out]
((-1)^(1/4)*Sqrt[1 - (-2)^(1/4)]*ArcTan[x/Sqrt[1 - (-2)^(1/4)]])/(4*2^(3/4)) - ((-1)^(3/4)*Sqrt[1 + I*(-2)^(1/
4)]*ArcTan[x/Sqrt[1 + I*(-2)^(1/4)]])/(4*2^(3/4)) - ((-1)^(1/4)*Sqrt[1 + (-2)^(1/4)]*ArcTan[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))]*ArcTan[Sqrt[(1 + I)/((1 +
I) + 2^(3/4))]*x]
________________________________________________________________________________________
Rubi [A] time = 0.276079, antiderivative size = 188, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{6740, 204, 203, 1972, 205} \[ \frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{1}{8} i \left (\sqrt [4]{-2}+\sqrt{2}\right ) \sqrt{\frac{1+i}{2^{3/4}+(1+i)}} \tan ^{-1}\left (\sqrt{\frac{1+i}{2^{3/4}+(1+i)}} x\right ) \]
Antiderivative was successfully verified.
[In]
Int[x^2/(2 + (1 + x^2)^4),x]
[Out]
((-1)^(1/4)*Sqrt[1 - (-2)^(1/4)]*ArcTan[x/Sqrt[1 - (-2)^(1/4)]])/(4*2^(3/4)) - ((-1)^(3/4)*Sqrt[1 + I*(-2)^(1/
4)]*ArcTan[x/Sqrt[1 + I*(-2)^(1/4)]])/(4*2^(3/4)) - ((-1)^(1/4)*Sqrt[1 + (-2)^(1/4)]*ArcTan[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))]*ArcTan[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 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 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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}} \tan ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tan ^{-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}} \tan ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tan ^{-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}}} \tan ^{-1}\left (\sqrt{\frac{1+i}{(1+i)+2^{3/4}}} x\right )\\ \end{align*}
Mathematica [C] time = 0.0134079, 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
________________________________________________________________________________________
Maple [C] time = 0.007, size = 54, 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 = 9.79067, size = 8759, normalized size = 46.59 \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(-122
88*(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(1/81
92*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(-1/819
2*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*sqrt(-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*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) + 12
8*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 - 122
88*(-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*sq
rt(-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*sqrt(2))) + 16384*(sqrt(2)*(I*sqrt(2) + 128*sqrt(-
1/8192*I*sqrt(2))) - sqrt(2))*(1/256*I*sqrt(2) - 1/2*sqrt(1/8192*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*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(1/81
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*sq
rt(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*(1/25
6*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 - 1228
8*(-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) - 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))) + 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) - 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*sqr
t(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))) + 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) - 2*I*sqrt(2) - 256*sqr
t(-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*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*sqrt(-1/8192*I*sqrt(2))) + x)
________________________________________________________________________________________
Sympy [A] time = 0.188685, 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)