3.389 \(\int \frac{x^2}{2+\left (1+x^2\right )^4} \, dx\)

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]

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Rubi [A]  time = 0.574583, 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 \[ \frac{\sqrt [4]{-1} \sqrt{1-\sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1-\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{(-1)^{3/4} \sqrt{1+i \sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{-1} \sqrt{1+\sqrt [4]{-2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{-2}}}\right )}{4\ 2^{3/4}}+\frac{1}{8} i \left (\sqrt [4]{-2}+\sqrt{2}\right ) \sqrt{\frac{1+i}{2^{3/4}+(1+i)}} \tan ^{-1}\left (\sqrt{\frac{1+i}{2^{3/4}+(1+i)}} x\right ) \]

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]

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Rubi in Sympy [A]  time = 83.5536, size = 236, normalized size = 1.26 \[ \frac{\left (\sqrt [4]{2} - 2 i + \sqrt [4]{2} i\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- 2^{\frac{3}{4}} + 2 - 2^{\frac{3}{4}} i}} \right )}}{8 \sqrt{- 2^{\frac{3}{4}} + 2 - 2^{\frac{3}{4}} i}} + \frac{\left (\sqrt [4]{2} - \sqrt [4]{2} i + 2 i\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- 2^{\frac{3}{4}} + 2 + 2^{\frac{3}{4}} i}} \right )}}{8 \sqrt{- 2^{\frac{3}{4}} + 2 + 2^{\frac{3}{4}} i}} - \frac{\left (\sqrt [4]{2} - 2 i - \sqrt [4]{2} i\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{2^{\frac{3}{4}} + 2 - 2^{\frac{3}{4}} i}} \right )}}{8 \sqrt{2^{\frac{3}{4}} + 2 - 2^{\frac{3}{4}} i}} - \frac{\left (\sqrt [4]{2} + \sqrt [4]{2} i + 2 i\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{2^{\frac{3}{4}} + 2 + 2^{\frac{3}{4}} i}} \right )}}{8 \sqrt{2^{\frac{3}{4}} + 2 + 2^{\frac{3}{4}} i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(2+(x**2+1)**4),x)

[Out]

(2**(1/4) - 2*I + 2**(1/4)*I)*atan(sqrt(2)*x/sqrt(-2**(3/4) + 2 - 2**(3/4)*I))/(
8*sqrt(-2**(3/4) + 2 - 2**(3/4)*I)) + (2**(1/4) - 2**(1/4)*I + 2*I)*atan(sqrt(2)
*x/sqrt(-2**(3/4) + 2 + 2**(3/4)*I))/(8*sqrt(-2**(3/4) + 2 + 2**(3/4)*I)) - (2**
(1/4) - 2*I - 2**(1/4)*I)*atan(sqrt(2)*x/sqrt(2**(3/4) + 2 - 2**(3/4)*I))/(8*sqr
t(2**(3/4) + 2 - 2**(3/4)*I)) - (2**(1/4) + 2**(1/4)*I + 2*I)*atan(sqrt(2)*x/sqr
t(2**(3/4) + 2 + 2**(3/4)*I))/(8*sqrt(2**(3/4) + 2 + 2**(3/4)*I))

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Mathematica [C]  time = 0.0188428, 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 = 0.009, size = 54, normalized size = 0.3 \[{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+4\,{{\it \_Z}}^{6}+6\,{{\it \_Z}}^{4}+4\,{{\it \_Z}}^{2}+3 \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}+3\,{{\it \_R}}^{5}+3\,{{\it \_R}}^{3}+{\it \_R}}}} \]

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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (x^{2} + 1\right )}^{4} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^2 + 1)^4 + 2),x, algorithm="maxima")

[Out]

integrate(x^2/((x^2 + 1)^4 + 2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^2 + 1)^4 + 2),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.591718, size = 39, normalized size = 0.21 \[ \operatorname{RootSum}{\left (1073741824 t^{8} + 65536 t^{4} + 1024 t^{2} + 3, \left ( t \mapsto t \log{\left (67108864 t^{7} - 262144 t^{5} + 4096 t^{3} + 40 t + x \right )} \right )\right )} \]

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(67108
864*_t**7 - 262144*_t**5 + 4096*_t**3 + 40*_t + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (x^{2} + 1\right )}^{4} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^2 + 1)^4 + 2),x, algorithm="giac")

[Out]

integrate(x^2/((x^2 + 1)^4 + 2), x)