3.387 \(\int \frac{x^2}{2-(1+x^2)^4} \, dx\)
Optimal. Leaf size=157 \[ \frac{i \sqrt{1-i \sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{i \sqrt{1+i \sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt{1+\sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt{\sqrt [4]{2}-1} \tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}} \]
[Out]
((I/4)*Sqrt[1 - I*2^(1/4)]*ArcTan[x/Sqrt[1 - I*2^(1/4)]])/2^(3/4) - ((I/4)*Sqrt[1 + I*2^(1/4)]*ArcTan[x/Sqrt[1
+ I*2^(1/4)]])/2^(3/4) - (Sqrt[1 + 2^(1/4)]*ArcTan[x/Sqrt[1 + 2^(1/4)]])/(4*2^(3/4)) + (Sqrt[-1 + 2^(1/4)]*Ar
cTanh[x/Sqrt[-1 + 2^(1/4)]])/(4*2^(3/4))
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Rubi [A] time = 0.168297, antiderivative size = 157, 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, 203, 1972, 205} \[ \frac{i \sqrt{1-i \sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{i \sqrt{1+i \sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt{1+\sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt{\sqrt [4]{2}-1} \tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
[In]
Int[x^2/(2 - (1 + x^2)^4),x]
[Out]
((I/4)*Sqrt[1 - I*2^(1/4)]*ArcTan[x/Sqrt[1 - I*2^(1/4)]])/2^(3/4) - ((I/4)*Sqrt[1 + I*2^(1/4)]*ArcTan[x/Sqrt[1
+ I*2^(1/4)]])/2^(3/4) - (Sqrt[1 + 2^(1/4)]*ArcTan[x/Sqrt[1 + 2^(1/4)]])/(4*2^(3/4)) + (Sqrt[-1 + 2^(1/4)]*Ar
cTanh[x/Sqrt[-1 + 2^(1/4)]])/(4*2^(3/4))
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 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}+\sqrt{2}}{8 \left (-1+\sqrt [4]{2}-x^2\right )}+\frac{-\sqrt [4]{2}-\sqrt{2}}{8 \left (1+\sqrt [4]{2}+x^2\right )}+\frac{-\sqrt [4]{2}-i \sqrt{2}}{8 \left (\sqrt [4]{2}-i \left (1+x^2\right )\right )}+\frac{-\sqrt [4]{2}+i \sqrt{2}}{8 \left (\sqrt [4]{2}+i \left (1+x^2\right )\right )}\right ) \, dx\\ &=-\frac{\left (1-\sqrt [4]{2}\right ) \int \frac{1}{-1+\sqrt [4]{2}-x^2} \, dx}{4\ 2^{3/4}}-\frac{\left (1-i \sqrt [4]{2}\right ) \int \frac{1}{\sqrt [4]{2}+i \left (1+x^2\right )} \, dx}{4\ 2^{3/4}}-\frac{\left (1+i \sqrt [4]{2}\right ) \int \frac{1}{\sqrt [4]{2}-i \left (1+x^2\right )} \, dx}{4\ 2^{3/4}}-\frac{\left (1+\sqrt [4]{2}\right ) \int \frac{1}{1+\sqrt [4]{2}+x^2} \, dx}{4\ 2^{3/4}}\\ &=-\frac{\sqrt{1+\sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt{-1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{\left (1-i \sqrt [4]{2}\right ) \int \frac{1}{i+\sqrt [4]{2}+i x^2} \, dx}{4\ 2^{3/4}}-\frac{\left (1+i \sqrt [4]{2}\right ) \int \frac{1}{-i+\sqrt [4]{2}-i x^2} \, dx}{4\ 2^{3/4}}\\ &=\frac{i \sqrt{1-i \sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{i \sqrt{1+i \sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{\sqrt{1+\sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt{-1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0167993, size = 61, normalized size = 0.39 \[ -\frac{1}{8} \text{RootSum}\left [\text{$\#$1}^8+4 \text{$\#$1}^6+6 \text{$\#$1}^4+4 \text{$\#$1}^2-1\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^6+3 \text{$\#$1}^4+3 \text{$\#$1}^2+1}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(2 - (1 + x^2)^4),x]
[Out]
-RootSum[-1 + 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 \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}-1 \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^2-1))
________________________________________________________________________________________
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)
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Fricas [B] time = 8.71275, size = 4887, normalized size = 31.13 \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(1/2*sqrt(2) + sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)
) - 3/16*(2^(3/4) - sqrt(2))^2 + 1))*log(1/4*((sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4) - sqrt(2))^2 +
sqrt(2)*(2^(3/4) + sqrt(2))^2 - (sqrt(2)*(2^(3/4) + sqrt(2))^2 - 4*sqrt(2))*(2^(3/4) - sqrt(2)) + 4*sqrt(-3/16
*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1)*((sqrt(
2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4) - sqrt(2)) - sqrt(2)*(2^(3/4) + sqrt(2)) - 4*sqrt(2)) - 4*sqrt(2)*(
2^(3/4) + sqrt(2)) + 4*sqrt(2))*sqrt(1/2*sqrt(2) + sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*
(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1)) + 6*x) + 1/16*sqrt(2)*sqrt(1/2*sqrt(2) + sqrt(-3/16*(2^
(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1))*log(-1/4*(
(sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4) - sqrt(2))^2 + sqrt(2)*(2^(3/4) + sqrt(2))^2 - (sqrt(2)*(2^(3
/4) + sqrt(2))^2 - 4*sqrt(2))*(2^(3/4) - sqrt(2)) + 4*sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2
))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1)*((sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4) - s
qrt(2)) - sqrt(2)*(2^(3/4) + sqrt(2)) - 4*sqrt(2)) - 4*sqrt(2)*(2^(3/4) + sqrt(2)) + 4*sqrt(2))*sqrt(1/2*sqrt(
2) + sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))
^2 + 1)) + 6*x) - 1/16*sqrt(2)*sqrt(1/2*sqrt(2) - sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(
2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1))*log(1/4*((sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4
) - sqrt(2))^2 + sqrt(2)*(2^(3/4) + sqrt(2))^2 - (sqrt(2)*(2^(3/4) + sqrt(2))^2 - 4*sqrt(2))*(2^(3/4) - sqrt(2
)) - 4*sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2
))^2 + 1)*((sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4) - sqrt(2)) - sqrt(2)*(2^(3/4) + sqrt(2)) - 4*sqrt(
2)) - 4*sqrt(2)*(2^(3/4) + sqrt(2)) + 4*sqrt(2))*sqrt(1/2*sqrt(2) - sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^
(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1)) + 6*x) + 1/16*sqrt(2)*sqrt(1/2*sqrt(2)
- sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2
+ 1))*log(-1/4*((sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt(2))*(2^(3/4) - sqrt(2))^2 + sqrt(2)*(2^(3/4) + sqrt(2))^2
- (sqrt(2)*(2^(3/4) + sqrt(2))^2 - 4*sqrt(2))*(2^(3/4) - sqrt(2)) - 4*sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*
(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2^(3/4) - sqrt(2))^2 + 1)*((sqrt(2)*(2^(3/4) + sqrt(2)) + sqrt
(2))*(2^(3/4) - sqrt(2)) - sqrt(2)*(2^(3/4) + sqrt(2)) - 4*sqrt(2)) - 4*sqrt(2)*(2^(3/4) + sqrt(2)) + 4*sqrt(2
))*sqrt(1/2*sqrt(2) - sqrt(-3/16*(2^(3/4) + sqrt(2))^2 + 1/8*(2^(3/4) + sqrt(2))*(2^(3/4) - sqrt(2)) - 3/16*(2
^(3/4) - sqrt(2))^2 + 1)) + 6*x) + 1/16*sqrt(2^(3/4) - sqrt(2))*log(1/4*((2^(3/4) + sqrt(2))^3 + (2^(3/4) + sq
rt(2) + 1)*(2^(3/4) - sqrt(2))^2 - ((2^(3/4) + sqrt(2))^2 - 4)*(2^(3/4) - sqrt(2)) - 4*2^(3/4) - 4*sqrt(2) - 6
)*sqrt(2^(3/4) - sqrt(2)) + 3*x) - 1/16*sqrt(2^(3/4) - sqrt(2))*log(-1/4*((2^(3/4) + sqrt(2))^3 + (2^(3/4) + s
qrt(2) + 1)*(2^(3/4) - sqrt(2))^2 - ((2^(3/4) + sqrt(2))^2 - 4)*(2^(3/4) - sqrt(2)) - 4*2^(3/4) - 4*sqrt(2) -
6)*sqrt(2^(3/4) - sqrt(2)) + 3*x) - sqrt(-1/256*2^(3/4) - 1/256*sqrt(2))*log(4*((2^(3/4) + sqrt(2))^3 - (2^(3/
4) + sqrt(2))^2 - 10)*sqrt(-1/256*2^(3/4) - 1/256*sqrt(2)) + 3*x) + sqrt(-1/256*2^(3/4) - 1/256*sqrt(2))*log(-
4*((2^(3/4) + sqrt(2))^3 - (2^(3/4) + sqrt(2))^2 - 10)*sqrt(-1/256*2^(3/4) - 1/256*sqrt(2)) + 3*x)
________________________________________________________________________________________
Sympy [A] time = 0.206149, size = 41, normalized size = 0.26 \begin{align*} - \operatorname{RootSum}{\left (1073741824 t^{8} - 65536 t^{4} + 1024 t^{2} - 1, \left ( t \mapsto t \log{\left (- \frac{67108864 t^{7}}{3} - \frac{262144 t^{5}}{3} - \frac{40 t}{3} + 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 - 1, Lambda(_t, _t*log(-67108864*_t**7/3 - 262144*_t**5/3
- 40*_t/3 + x)))
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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)