3.388 \(\int \frac{x^2}{2-(1-x^2)^4} \, dx\)

Optimal. Leaf size=157 \[ -\frac{\sqrt{\sqrt [4]{2}-1} \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}}-\frac{i \sqrt{1-i \sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{i \sqrt{1+i \sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt{1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}} \]

[Out]

-(Sqrt[-1 + 2^(1/4)]*ArcTan[x/Sqrt[-1 + 2^(1/4)]])/(4*2^(3/4)) - ((I/4)*Sqrt[1 - I*2^(1/4)]*ArcTanh[x/Sqrt[1 -
I*2^(1/4)]])/2^(3/4) + ((I/4)*Sqrt[1 + I*2^(1/4)]*ArcTanh[x/Sqrt[1 + I*2^(1/4)]])/2^(3/4) + (Sqrt[1 + 2^(1/4)
]*ArcTanh[x/Sqrt[1 + 2^(1/4)]])/(4*2^(3/4))

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Rubi [A]  time = 0.111525, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6740, 206, 203, 1972, 208} \[ -\frac{\sqrt{\sqrt [4]{2}-1} \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}}-\frac{i \sqrt{1-i \sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{i \sqrt{1+i \sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{\sqrt{1+\sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(2 - (1 - x^2)^4),x]

[Out]

-(Sqrt[-1 + 2^(1/4)]*ArcTan[x/Sqrt[-1 + 2^(1/4)]])/(4*2^(3/4)) - ((I/4)*Sqrt[1 - I*2^(1/4)]*ArcTanh[x/Sqrt[1 -
I*2^(1/4)]])/2^(3/4) + ((I/4)*Sqrt[1 + I*2^(1/4)]*ArcTanh[x/Sqrt[1 + I*2^(1/4)]])/2^(3/4) + (Sqrt[1 + 2^(1/4)
]*ArcTanh[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 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}+\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{\sqrt{-1+\sqrt [4]{2}} \tan ^{-1}\left (\frac{x}{\sqrt{-1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac{i \sqrt{1-i \sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac{i \sqrt{1+i \sqrt [4]{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{1+i \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 = 0.0178763, 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 = 56, normalized size = 0.4 \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))

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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 = 8.6147, size = 4898, normalized size = 31.2 \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(2)
*(2^(3/4) - sqrt(2)) - sqrt(2))*(2^(3/4) + sqrt(2)) + sqrt(2)*(2^(3/4) - sqrt(2)) - 4*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) - 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(2)*(2^(3/4) - sqrt(2)) - sqrt(2))*(2^(3/4) + s
qrt(2)) + sqrt(2)*(2^(3/4) - sqrt(2)) - 4*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) - 4*sqrt(2)*(2^(3/4) - sqrt(2)) - 4*sqrt(2))*sqrt(-1/2*s
qrt(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) + s
qrt(2)) - 4*((sqrt(2)*(2^(3/4) - sqrt(2)) - sqrt(2))*(2^(3/4) + sqrt(2)) + sqrt(2)*(2^(3/4) - sqrt(2)) - 4*sqr
t(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) - 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) - sqr
t(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) - sqr
t(2))^2 - (sqrt(2)*(2^(3/4) - sqrt(2))^2 - 4*sqrt(2))*(2^(3/4) + sqrt(2)) - 4*((sqrt(2)*(2^(3/4) - sqrt(2)) -
sqrt(2))*(2^(3/4) + sqrt(2)) + sqrt(2)*(2^(3/4) - sqrt(2)) - 4*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) - 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) + sqrt(2))^2*(2^(3/4) - sqrt(2) - 1) - ((2^(3/4) - sqrt(2))^2 - 4)*(2^(3/4) + sqrt(2)) - 4*2^(3/4) + 4*sqr
t(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) + sqrt(2))^2*(2^(3/4) - sqrt(2) - 1) - ((2^(3/4) - sqrt(2))^2 - 4)*(2^(3/4) + sqrt(2)) - 4*2^(3/4) + 4*sq
rt(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)

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Sympy [A]  time = 0.20163, 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)