\(\int \frac {x^2}{2-(1-x^2)^4} \, dx\) [388]

Optimal result

Integrand size = 19, antiderivative size = 157 \[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=-\frac {\sqrt {-1+\sqrt [4]{2}} \arctan \left (\frac {x}{\sqrt {-1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}}-\frac {i \sqrt {1-i \sqrt [4]{2}} \text {arctanh}\left (\frac {x}{\sqrt {1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {i \sqrt {1+i \sqrt [4]{2}} \text {arctanh}\left (\frac {x}{\sqrt {1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\sqrt {1+\sqrt [4]{2}} \text {arctanh}\left (\frac {x}{\sqrt {1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}} \]

[Out]

-1/8*I*arctanh(x/(1-I*2^(1/4))^(1/2))*(1-I*2^(1/4))^(1/2)*2^(1/4)+1/8*I*arctanh(x/(1+I*2^(1/4))^(1/2))*(1+I*2^
(1/4))^(1/2)*2^(1/4)-1/8*arctan(x/(-1+2^(1/4))^(1/2))*(-1+2^(1/4))^(1/2)*2^(1/4)+1/8*arctanh(x/(1+2^(1/4))^(1/
2))*(1+2^(1/4))^(1/2)*2^(1/4)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6872, 212, 209, 1997, 214} \[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=-\frac {\sqrt {\sqrt [4]{2}-1} \arctan \left (\frac {x}{\sqrt {\sqrt [4]{2}-1}}\right )}{4\ 2^{3/4}}-\frac {i \sqrt {1-i \sqrt [4]{2}} \text {arctanh}\left (\frac {x}{\sqrt {1-i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {i \sqrt {1+i \sqrt [4]{2}} \text {arctanh}\left (\frac {x}{\sqrt {1+i \sqrt [4]{2}}}\right )}{4\ 2^{3/4}}+\frac {\sqrt {1+\sqrt [4]{2}} \text {arctanh}\left (\frac {x}{\sqrt {1+\sqrt [4]{2}}}\right )}{4\ 2^{3/4}} \]

[In]

Int[x^2/(2 - (1 - x^2)^4),x]

[Out]

-1/4*(Sqrt[-1 + 2^(1/4)]*ArcTan[x/Sqrt[-1 + 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) + ((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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1997

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] &&  !BinomialMatchQ[
u, x]

Rule 6872

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.39 \[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=-\frac {1}{8} \text {RootSum}\left [-1-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+3 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]

[In]

Integrate[x^2/(2 - (1 - x^2)^4),x]

[Out]

-1/8*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) & ]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.36

method	result	size
default	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+3 \textit {\_R}^{3}-\textit {\_R}}\right )}{8}\)	\(56\)
risch	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+3 \textit {\_R}^{3}-\textit {\_R}}\right )}{8}\)	\(56\)

[In]

int(x^2/(2-(-x^2+1)^4),x,method=_RETURNVERBOSE)

[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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (97) = 194\).

Time = 0.99 (sec) , antiderivative size = 1546, normalized size of antiderivative = 9.85 \[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=\text {Too large to display} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.26 \[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=- \operatorname {RootSum} {\left (1073741824 t^{8} - 65536 t^{4} - 1024 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {67108864 t^{7}}{3} + \frac {262144 t^{5}}{3} + \frac {40 t}{3} + x \right )} \right )\right )} \]

[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)))

Maxima [F]

\[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=\int { -\frac {x^{2}}{{\left (x^{2} - 1\right )}^{4} - 2} \,d x } \]

[In]

integrate(x^2/(2-(-x^2+1)^4),x, algorithm="maxima")

[Out]

-integrate(x^2/((x^2 - 1)^4 - 2), x)

Giac [F]

\[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=\int { -\frac {x^{2}}{{\left (x^{2} - 1\right )}^{4} - 2} \,d x } \]

[In]

integrate(x^2/(2-(-x^2+1)^4),x, algorithm="giac")

[Out]

integrate(-x^2/((x^2 - 1)^4 - 2), x)

Mupad [B] (verification not implemented)

Time = 9.94 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{2-\left (1-x^2\right )^4} \, dx=\sum _{k=1}^8\ln \left (-\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )\,\left (56\,x+\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )\,\left (\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )\,\left (4096\,x+{\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )}^2\,\left (262144\,x-{\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right )}^2\,x\,67108864\right )\right )+256\right )\right )-1\right )\,\mathrm {root}\left (z^8-\frac {z^4}{16384}-\frac {z^2}{1048576}-\frac {1}{1073741824},z,k\right ) \]

[In]

int(-x^2/((x^2 - 1)^4 - 2),x)

[Out]

symsum(log(- root(z^8 - z^4/16384 - z^2/1048576 - 1/1073741824, z, k)*(56*x + root(z^8 - z^4/16384 - z^2/10485
76 - 1/1073741824, z, k)*(root(z^8 - z^4/16384 - z^2/1048576 - 1/1073741824, z, k)*(4096*x + root(z^8 - z^4/16
384 - z^2/1048576 - 1/1073741824, z, k)^2*(262144*x - 67108864*root(z^8 - z^4/16384 - z^2/1048576 - 1/10737418
24, z, k)^2*x)) + 256)) - 1)*root(z^8 - z^4/16384 - z^2/1048576 - 1/1073741824, z, k), k, 1, 8)