\(\int \frac {x^{10}}{1+2 x^4+x^8} \, dx\) [31]

Optimal result

Integrand size = 16, antiderivative size = 85 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {x^3}{3}+\frac {x^3}{4 \left (1+x^4\right )}+\frac {7 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {7 \arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {7 \text {arctanh}\left (\frac {\sqrt {2} x}{1+x^2}\right )}{8 \sqrt {2}} \] Output:

1/3*x^3+x^3/(4*x^4+4)-7/16*arctan(-1+x*2^(1/2))*2^(1/2)-7/16*arctan(1+x*2^ 
(1/2))*2^(1/2)+7/16*arctanh(2^(1/2)*x/(x^2+1))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.15 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {1}{96} \left (32 x^3+\frac {24 x^3}{1+x^4}+42 \sqrt {2} \arctan \left (1-\sqrt {2} x\right )-42 \sqrt {2} \arctan \left (1+\sqrt {2} x\right )-21 \sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )+21 \sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )\right ) \] Input:

Integrate[x^10/(1 + 2*x^4 + x^8),x]
 

Output:

(32*x^3 + (24*x^3)/(1 + x^4) + 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 42*Sqrt[ 
2]*ArcTan[1 + Sqrt[2]*x] - 21*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + 21*Sqrt[2 
]*Log[1 + Sqrt[2]*x + x^2])/96
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1380, 817, 843, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^{10}}{x^8+2 x^4+1} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle \int \frac {x^{10}}{\left (x^4+1\right )^2}dx\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {7}{4} \int \frac {x^6}{x^4+1}dx-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {7}{4} \left (\frac {x^3}{3}-\int \frac {x^2}{x^4+1}dx\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx-\frac {1}{2} \int \frac {x^2+1}{x^4+1}dx+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{x^2-\sqrt {2} x+1}dx-\frac {1}{2} \int \frac {1}{x^2+\sqrt {2} x+1}dx\right )+\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx+\frac {1}{2} \left (\frac {\int \frac {1}{-\left (\sqrt {2} x+1\right )^2-1}d\left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\left (1-\sqrt {2} x\right )^2-1}d\left (1-\sqrt {2} x\right )}{\sqrt {2}}\right )+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx+\frac {1}{2} \left (\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}\right )+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 x}{x^2-\sqrt {2} x+1}dx}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} x+1\right )}{x^2+\sqrt {2} x+1}dx}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}\right )+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 x}{x^2-\sqrt {2} x+1}dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} x+1\right )}{x^2+\sqrt {2} x+1}dx}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}\right )+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 x}{x^2-\sqrt {2} x+1}dx}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} x+1}{x^2+\sqrt {2} x+1}dx\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}\right )+\frac {x^3}{3}\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {7}{4} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}\right )+\frac {x^3}{3}+\frac {1}{2} \left (\frac {\log \left (x^2+\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^2-\sqrt {2} x+1\right )}{2 \sqrt {2}}\right )\right )-\frac {x^7}{4 \left (x^4+1\right )}\)

Input:

Int[x^10/(1 + 2*x^4 + x^8),x]
 

Output:

-1/4*x^7/(1 + x^4) + (7*(x^3/3 + (ArcTan[1 - Sqrt[2]*x]/Sqrt[2] - ArcTan[1 
 + Sqrt[2]*x]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*x + x^2]/Sqrt[2] + Log[1 
+ Sqrt[2]*x + x^2]/(2*Sqrt[2]))/2))/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 817

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( 
n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n 
*((m - n + 1)/(b*n*(p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x 
] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  ! 
ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 843

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n 
 - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.47

method	result	size
risch	\(\frac {x^{3}}{3}+\frac {x^{3}}{4 x^{4}+4}-\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{16}\)	\(40\)
default	\(\frac {x^{3}}{3}+\frac {x^{3}}{4 x^{4}+4}-\frac {7 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-x \sqrt {2}+1}{x^{2}+x \sqrt {2}+1}\right )+2 \arctan \left (1+x \sqrt {2}\right )+2 \arctan \left (-1+x \sqrt {2}\right )\right )}{32}\)	\(70\)

Input:

int(x^10/(x^8+2*x^4+1),x,method=_RETURNVERBOSE)
 

Output:

1/3*x^3+1/4*x^3/(x^4+1)-7/16*sum(1/_R*ln(x-_R),_R=RootOf(_Z^4+1))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {32 \, x^{7} + 56 \, x^{3} - 42 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (\sqrt {2} x + 1\right ) - 42 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (\sqrt {2} x - 1\right ) + 21 \, \sqrt {2} {\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt {2} x + 1\right ) - 21 \, \sqrt {2} {\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt {2} x + 1\right )}{96 \, {\left (x^{4} + 1\right )}} \] Input:

integrate(x^10/(x^8+2*x^4+1),x, algorithm="fricas")
 

Output:

1/96*(32*x^7 + 56*x^3 - 42*sqrt(2)*(x^4 + 1)*arctan(sqrt(2)*x + 1) - 42*sq 
rt(2)*(x^4 + 1)*arctan(sqrt(2)*x - 1) + 21*sqrt(2)*(x^4 + 1)*log(x^2 + sqr 
t(2)*x + 1) - 21*sqrt(2)*(x^4 + 1)*log(x^2 - sqrt(2)*x + 1))/(x^4 + 1)
 

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {x^{3}}{3} + \frac {x^{3}}{4 x^{4} + 4} - \frac {7 \sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} + \frac {7 \sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} - \frac {7 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} - \frac {7 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} \] Input:

integrate(x**10/(x**8+2*x**4+1),x)
 

Output:

x**3/3 + x**3/(4*x**4 + 4) - 7*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + 7*sq 
rt(2)*log(x**2 + sqrt(2)*x + 1)/32 - 7*sqrt(2)*atan(sqrt(2)*x - 1)/16 - 7* 
sqrt(2)*atan(sqrt(2)*x + 1)/16
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {1}{3} \, x^{3} + \frac {x^{3}}{4 \, {\left (x^{4} + 1\right )}} - \frac {7}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {7}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {7}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {7}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \] Input:

integrate(x^10/(x^8+2*x^4+1),x, algorithm="maxima")
 

Output:

1/3*x^3 + 1/4*x^3/(x^4 + 1) - 7/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt( 
2))) - 7/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) + 7/32*sqrt(2)*log 
(x^2 + sqrt(2)*x + 1) - 7/32*sqrt(2)*log(x^2 - sqrt(2)*x + 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {1}{3} \, x^{3} + \frac {x^{3}}{4 \, {\left (x^{4} + 1\right )}} - \frac {7}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {7}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {7}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {7}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \] Input:

integrate(x^10/(x^8+2*x^4+1),x, algorithm="giac")
 

Output:

1/3*x^3 + 1/4*x^3/(x^4 + 1) - 7/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt( 
2))) - 7/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) + 7/32*sqrt(2)*log 
(x^2 + sqrt(2)*x + 1) - 7/32*sqrt(2)*log(x^2 - sqrt(2)*x + 1)
 

Mupad [B] (verification not implemented)

Time = 20.82 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.60 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {x^3}{4\,\left (x^4+1\right )}+\frac {x^3}{3}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {7}{16}+\frac {7}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {7}{16}-\frac {7}{16}{}\mathrm {i}\right ) \] Input:

int(x^10/(2*x^4 + x^8 + 1),x)
 

Output:

x^3/(4*(x^4 + 1)) - 2^(1/2)*atan(2^(1/2)*x*(1/2 + 1i/2))*(7/16 + 7i/16) - 
2^(1/2)*atan(2^(1/2)*x*(1/2 - 1i/2))*(7/16 - 7i/16) + x^3/3
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.82 \[ \int \frac {x^{10}}{1+2 x^4+x^8} \, dx=\frac {42 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right ) x^{4}+42 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right )-42 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right ) x^{4}-42 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right )-21 \sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, x +x^{2}+1\right ) x^{4}-21 \sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, x +x^{2}+1\right )+21 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, x +x^{2}+1\right ) x^{4}+21 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, x +x^{2}+1\right )+32 x^{7}+56 x^{3}}{96 x^{4}+96} \] Input:

int(x^10/(x^8+2*x^4+1),x)
 

Output:

(42*sqrt(2)*atan((sqrt(2) - 2*x)/sqrt(2))*x**4 + 42*sqrt(2)*atan((sqrt(2) 
- 2*x)/sqrt(2)) - 42*sqrt(2)*atan((sqrt(2) + 2*x)/sqrt(2))*x**4 - 42*sqrt( 
2)*atan((sqrt(2) + 2*x)/sqrt(2)) - 21*sqrt(2)*log( - sqrt(2)*x + x**2 + 1) 
*x**4 - 21*sqrt(2)*log( - sqrt(2)*x + x**2 + 1) + 21*sqrt(2)*log(sqrt(2)*x 
 + x**2 + 1)*x**4 + 21*sqrt(2)*log(sqrt(2)*x + x**2 + 1) + 32*x**7 + 56*x* 
*3)/(96*(x**4 + 1))