Integrand size = 17, antiderivative size = 219 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=4 x-\frac {x^3}{3}+\frac {1}{16} \sqrt {95+81 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {\arctan \left (\sqrt {-1+\sqrt {2}} x\right )}{8 \sqrt {2 \left (-1393+985 \sqrt {2}\right )}}-\frac {1}{16} \sqrt {95+81 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\text {arctanh}\left (\sqrt {1+\sqrt {2}} x\right )}{8 \sqrt {2 \left (1393+985 \sqrt {2}\right )}}+\frac {1}{16} \sqrt {-95+81 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )} x}{\sqrt {3}+x^2}\right ) \] Output:
4*x-1/3*x^3+1/16*(95+81*3^(1/2))^(1/2)*arctan(((-2+2*3^(1/2))^(1/2)-2*x)/( 2+2*3^(1/2))^(1/2))-1/8*arctan((2^(1/2)-1)^(1/2)*x)/(-2786+1970*2^(1/2))^( 1/2)-1/16*(95+81*3^(1/2))^(1/2)*arctan(((-2+2*3^(1/2))^(1/2)+2*x)/(2+2*3^( 1/2))^(1/2))+1/8*arctanh((1+2^(1/2))^(1/2)*x)/(2786+1970*2^(1/2))^(1/2)+1/ 16*(-95+81*3^(1/2))^(1/2)*arctanh((-2+2*3^(1/2))^(1/2)*x/(3^(1/2)+x^2))
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=4 x-\frac {x^3}{3}-\frac {\left (-i+11 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{8 \sqrt {2-2 i \sqrt {2}}}-\frac {\left (i+11 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{8 \sqrt {2+2 i \sqrt {2}}}-\frac {\left (41+29 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\left (-41+29 \sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (-1+\sqrt {2}\right )}} \] Input:
Integrate[x^10/(4 - (1 + x^2)^4),x]
Output:
4*x - x^3/3 - ((-I + 11*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(8*Sqrt[2 - (2*I)*Sqrt[2]]) - ((I + 11*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(8*Sq rt[2 + (2*I)*Sqrt[2]]) - ((41 + 29*Sqrt[2])*ArcTan[x/Sqrt[1 + Sqrt[2]]])/( 8*Sqrt[2*(1 + Sqrt[2])]) + ((-41 + 29*Sqrt[2])*ArcTanh[x/Sqrt[-1 + Sqrt[2] ]])/(8*Sqrt[2*(-1 + Sqrt[2])])
Time = 0.95 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2460, 2009}
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.
| \(\displaystyle \int \frac {x^{10}}{4-\left (x^2+1\right )^4} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (-x^2+\frac {12-29 x^2}{4 \left (x^4+2 x^2-1\right )}+\frac {-11 x^2-12}{4 \left (x^4+2 x^2+3\right )}+4\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} \sqrt {95+81 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (1393+985 \sqrt {2}\right )} \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )-\frac {1}{16} \sqrt {95+81 \sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (985 \sqrt {2}-1393\right )} \text {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )-\frac {x^3}{3}-\frac {1}{32} \sqrt {81 \sqrt {3}-95} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {81 \sqrt {3}-95} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+4 x\) |
Input:
Int[x^10/(4 - (1 + x^2)^4),x]
Output:
4*x - x^3/3 + (Sqrt[95 + 81*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x) /Sqrt[2*(1 + Sqrt[3])]])/16 - (Sqrt[(1393 + 985*Sqrt[2])/2]*ArcTan[x/Sqrt[ 1 + Sqrt[2]]])/8 - (Sqrt[95 + 81*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + (Sqrt[(-1393 + 985*Sqrt[2])/2]*ArcTanh[ x/Sqrt[-1 + Sqrt[2]]])/8 - (Sqrt[-95 + 81*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(- 1 + Sqrt[3])]*x + x^2])/32 + (Sqrt[-95 + 81*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2* (-1 + Sqrt[3])]*x + x^2])/32
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(-\frac {x^{3}}{3}+4 x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+380 \textit {\_Z}^{2}+19683\right )}{\sum }\textit {\_R} \ln \left (-14 \textit {\_R}^{3}-1249 \textit {\_R} +5913 x \right )\right )}{16}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+5572 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (34 \textit {\_R}^{3}+47321 \textit {\_R} +985 x \right )\right )}{16}\) | \(76\) |
| default | \(-\frac {x^{3}}{3}+4 x -\frac {\left (7 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-\sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}-x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{64}-\frac {\left (16 \sqrt {3}+\frac {\left (7 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-\sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (7 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-\sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{64}+\frac {\left (-16 \sqrt {3}-\frac {\left (7 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-\sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {3}}+2 x}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (-41+29 \sqrt {2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {\sqrt {2}-1}}-\frac {\left (41+29 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {1+\sqrt {2}}}\) | \(321\) |
Input:
int(x^10/(4-(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
-1/3*x^3+4*x+1/16*sum(_R*ln(-14*_R^3-1249*_R+5913*x),_R=RootOf(4*_Z^4+380* _Z^2+19683))+1/16*sum(_R*ln(34*_R^3+47321*_R+985*x),_R=RootOf(4*_Z^4+5572* _Z^2-1))
Time = 0.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.30 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=-\frac {1}{3} \, x^{3} + \frac {1}{16} \, \sqrt {81 \, \sqrt {3} + 95} \arctan \left (\frac {1}{146} \, {\left (8 \, \sqrt {3} x + \sqrt {81 \, \sqrt {3} - 95} {\left (\sqrt {3} - 1\right )} - 22 \, x\right )} \sqrt {81 \, \sqrt {3} + 95}\right ) - \frac {1}{16} \, \sqrt {81 \, \sqrt {3} + 95} \arctan \left (-\frac {1}{146} \, {\left (8 \, \sqrt {3} x - \sqrt {81 \, \sqrt {3} - 95} {\left (\sqrt {3} - 1\right )} - 22 \, x\right )} \sqrt {81 \, \sqrt {3} + 95}\right ) + \frac {1}{8} \, \sqrt {\frac {985}{2} \, \sqrt {2} + \frac {1393}{2}} \arctan \left ({\left (41 \, \sqrt {2} x - 58 \, x\right )} \sqrt {\frac {985}{2} \, \sqrt {2} + \frac {1393}{2}}\right ) + \frac {1}{32} \, \sqrt {81 \, \sqrt {3} - 95} \log \left (73 \, x^{2} + {\left (7 \, \sqrt {3} x + x\right )} \sqrt {81 \, \sqrt {3} - 95} + 73 \, \sqrt {3}\right ) - \frac {1}{32} \, \sqrt {81 \, \sqrt {3} - 95} \log \left (73 \, x^{2} - {\left (7 \, \sqrt {3} x + x\right )} \sqrt {81 \, \sqrt {3} - 95} + 73 \, \sqrt {3}\right ) + \frac {1}{16} \, \sqrt {\frac {985}{2} \, \sqrt {2} - \frac {1393}{2}} \log \left (\sqrt {\frac {985}{2} \, \sqrt {2} - \frac {1393}{2}} {\left (17 \, \sqrt {2} + 24\right )} + x\right ) - \frac {1}{16} \, \sqrt {\frac {985}{2} \, \sqrt {2} - \frac {1393}{2}} \log \left (-\sqrt {\frac {985}{2} \, \sqrt {2} - \frac {1393}{2}} {\left (17 \, \sqrt {2} + 24\right )} + x\right ) + 4 \, x \] Input:
integrate(x^10/(4-(x^2+1)^4),x, algorithm="fricas")
Output:
-1/3*x^3 + 1/16*sqrt(81*sqrt(3) + 95)*arctan(1/146*(8*sqrt(3)*x + sqrt(81* sqrt(3) - 95)*(sqrt(3) - 1) - 22*x)*sqrt(81*sqrt(3) + 95)) - 1/16*sqrt(81* sqrt(3) + 95)*arctan(-1/146*(8*sqrt(3)*x - sqrt(81*sqrt(3) - 95)*(sqrt(3) - 1) - 22*x)*sqrt(81*sqrt(3) + 95)) + 1/8*sqrt(985/2*sqrt(2) + 1393/2)*arc tan((41*sqrt(2)*x - 58*x)*sqrt(985/2*sqrt(2) + 1393/2)) + 1/32*sqrt(81*sqr t(3) - 95)*log(73*x^2 + (7*sqrt(3)*x + x)*sqrt(81*sqrt(3) - 95) + 73*sqrt( 3)) - 1/32*sqrt(81*sqrt(3) - 95)*log(73*x^2 - (7*sqrt(3)*x + x)*sqrt(81*sq rt(3) - 95) + 73*sqrt(3)) + 1/16*sqrt(985/2*sqrt(2) - 1393/2)*log(sqrt(985 /2*sqrt(2) - 1393/2)*(17*sqrt(2) + 24) + x) - 1/16*sqrt(985/2*sqrt(2) - 13 93/2)*log(-sqrt(985/2*sqrt(2) - 1393/2)*(17*sqrt(2) + 24) + x) + 4*x
Time = 1.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.42 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=- \frac {x^{3}}{3} + 4 x - \operatorname {RootSum} {\left (262144 t^{4} + 97280 t^{2} + 19683, \left ( t \mapsto t \log {\left (- \frac {672644552569389056 t^{7}}{355901897526345} - \frac {1303034319856467968 t^{5}}{118633965842115} - \frac {1405067195890829312 t^{3}}{355901897526345} - \frac {273568734602968628 t}{355901897526345} + x \right )} \right )\right )} - \operatorname {RootSum} {\left (262144 t^{4} + 1426432 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {672644552569389056 t^{7}}{355901897526345} - \frac {1303034319856467968 t^{5}}{118633965842115} - \frac {1405067195890829312 t^{3}}{355901897526345} - \frac {273568734602968628 t}{355901897526345} + x \right )} \right )\right )} \] Input:
integrate(x**10/(4-(x**2+1)**4),x)
Output:
-x**3/3 + 4*x - RootSum(262144*_t**4 + 97280*_t**2 + 19683, Lambda(_t, _t* log(-672644552569389056*_t**7/355901897526345 - 1303034319856467968*_t**5/ 118633965842115 - 1405067195890829312*_t**3/355901897526345 - 273568734602 968628*_t/355901897526345 + x))) - RootSum(262144*_t**4 + 1426432*_t**2 - 1, Lambda(_t, _t*log(-672644552569389056*_t**7/355901897526345 - 130303431 9856467968*_t**5/118633965842115 - 1405067195890829312*_t**3/3559018975263 45 - 273568734602968628*_t/355901897526345 + x)))
\[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=\int { -\frac {x^{10}}{{\left (x^{2} + 1\right )}^{4} - 4} \,d x } \] Input:
integrate(x^10/(4-(x^2+1)^4),x, algorithm="maxima")
Output:
-1/3*x^3 + 4*x - 1/4*integrate((29*x^2 - 12)/(x^4 + 2*x^2 - 1), x) - 1/4*i ntegrate((11*x^2 + 12)/(x^4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (156) = 312\).
Time = 0.46 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.83 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(x^10/(4-(x^2+1)^4),x, algorithm="giac")
Output:
-1/3*x^3 + 1/10368*sqrt(2)*(11*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 19 8*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 198*3^(3/4)*(sqrt(3 ) + 3)*sqrt(-6*sqrt(3) + 18) - 11*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 432*3^ (1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 432*3^(1/4)*sqrt(-6*sqrt(3) + 18))*ar ctan(1/3*3^(3/4)*(x + 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1/10368*sqrt(2)*(11*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 198* 3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 198*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 11*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 432*3^(1 /4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 432*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arct an(1/3*3^(3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1 /2)) + 1/20736*sqrt(2)*(198*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 11*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) - 11*3^(3/4)*(6*sqrt(3) + 18)^(3/2) - 198*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 432*3^(1/4 )*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 432*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^ 2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 1/20736*sqrt(2)*(198 *3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 11*3^(3/4)*sqrt(2)* (-6*sqrt(3) + 18)^(3/2) - 11*3^(3/4)*(6*sqrt(3) + 18)^(3/2) - 198*3^(3/4)* sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 432*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 432*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*sqrt(-1/6*s qrt(3) + 1/2) + sqrt(3)) - 1/16*sqrt(1970*sqrt(2) + 2786)*arctan(x/sqrt...
Time = 10.03 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.19 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=4\,x-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-190-\sqrt {2}\,146{}\mathrm {i}}\,16637138{}\mathrm {i}}{370131024+\sqrt {2}\,28616730{}\mathrm {i}}-\frac {2056994\,\sqrt {2}\,x\,\sqrt {-190-\sqrt {2}\,146{}\mathrm {i}}}{370131024+\sqrt {2}\,28616730{}\mathrm {i}}\right )\,\sqrt {-190-\sqrt {2}\,146{}\mathrm {i}}\,1{}\mathrm {i}}{16}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-190+\sqrt {2}\,146{}\mathrm {i}}\,16637138{}\mathrm {i}}{-370131024+\sqrt {2}\,28616730{}\mathrm {i}}+\frac {2056994\,\sqrt {2}\,x\,\sqrt {-190+\sqrt {2}\,146{}\mathrm {i}}}{-370131024+\sqrt {2}\,28616730{}\mathrm {i}}\right )\,\sqrt {-190+\sqrt {2}\,146{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {x^3}{3}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-1970\,\sqrt {2}-2786}\,1594003830{}\mathrm {i}}{128691674310\,\sqrt {2}+181998367260}+\frac {\sqrt {2}\,x\,\sqrt {-1970\,\sqrt {2}-2786}\,1092026160{}\mathrm {i}}{128691674310\,\sqrt {2}+181998367260}\right )\,\sqrt {-1970\,\sqrt {2}-2786}\,1{}\mathrm {i}}{16}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {1970\,\sqrt {2}-2786}\,1594003830{}\mathrm {i}}{128691674310\,\sqrt {2}-181998367260}-\frac {\sqrt {2}\,x\,\sqrt {1970\,\sqrt {2}-2786}\,1092026160{}\mathrm {i}}{128691674310\,\sqrt {2}-181998367260}\right )\,\sqrt {1970\,\sqrt {2}-2786}\,1{}\mathrm {i}}{16} \] Input:
int(-x^10/((x^2 + 1)^4 - 4),x)
Output:
4*x - (atan((x*(- 2^(1/2)*146i - 190)^(1/2)*16637138i)/(2^(1/2)*28616730i + 370131024) - (2056994*2^(1/2)*x*(- 2^(1/2)*146i - 190)^(1/2))/(2^(1/2)*2 8616730i + 370131024))*(- 2^(1/2)*146i - 190)^(1/2)*1i)/16 + (atan((x*(2^( 1/2)*146i - 190)^(1/2)*16637138i)/(2^(1/2)*28616730i - 370131024) + (20569 94*2^(1/2)*x*(2^(1/2)*146i - 190)^(1/2))/(2^(1/2)*28616730i - 370131024))* (2^(1/2)*146i - 190)^(1/2)*1i)/16 - x^3/3 - (atan((x*(- 1970*2^(1/2) - 278 6)^(1/2)*1594003830i)/(128691674310*2^(1/2) + 181998367260) + (2^(1/2)*x*( - 1970*2^(1/2) - 2786)^(1/2)*1092026160i)/(128691674310*2^(1/2) + 18199836 7260))*(- 1970*2^(1/2) - 2786)^(1/2)*1i)/16 + (atan((x*(1970*2^(1/2) - 278 6)^(1/2)*1594003830i)/(128691674310*2^(1/2) - 181998367260) - (2^(1/2)*x*( 1970*2^(1/2) - 2786)^(1/2)*1092026160i)/(128691674310*2^(1/2) - 1819983672 60))*(1970*2^(1/2) - 2786)^(1/2)*1i)/16
Time = 0.32 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.60 \[ \int \frac {x^{10}}{4-\left (1+x^2\right )^4} \, dx=\frac {7 \sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}+\frac {\sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {7 \sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {\sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {17 \sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{16}-\frac {3 \sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{2}-\frac {7 \sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {7 \sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {17 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{32}+\frac {17 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{32}+\frac {3 \sqrt {\sqrt {2}-1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{4}-\frac {3 \sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{4}-\frac {x^{3}}{3}+4 x \] Input:
int(x^10/(4-(x^2+1)^4),x)
Output:
(42*sqrt(sqrt(3) + 1)*sqrt(6)*atan((sqrt(sqrt(3) - 1)*sqrt(2) - 2*x)/(sqrt (sqrt(3) + 1)*sqrt(2))) + 6*sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt(3) - 1)*sqrt(2) - 2*x)/(sqrt(sqrt(3) + 1)*sqrt(2))) - 42*sqrt(sqrt(3) + 1)*sqr t(6)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/(sqrt(sqrt(3) + 1)*sqrt(2))) - 6*sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/(sqrt( sqrt(3) + 1)*sqrt(2))) - 204*sqrt(sqrt(2) + 1)*sqrt(2)*atan(x/sqrt(sqrt(2) + 1)) - 288*sqrt(sqrt(2) + 1)*atan(x/sqrt(sqrt(2) + 1)) - 21*sqrt(sqrt(3) - 1)*sqrt(6)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + 21*sq rt(sqrt(3) - 1)*sqrt(6)*log(sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + 3*sqrt(sqrt(3) - 1)*sqrt(2)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) - 3*sqrt(sqrt(3) - 1)*sqrt(2)*log(sqrt(sqrt(3) - 1)*sqrt(2)*x + s qrt(3) + x**2) - 102*sqrt(sqrt(2) - 1)*sqrt(2)*log( - sqrt(sqrt(2) - 1) + x) + 102*sqrt(sqrt(2) - 1)*sqrt(2)*log(sqrt(sqrt(2) - 1) + x) + 144*sqrt(s qrt(2) - 1)*log( - sqrt(sqrt(2) - 1) + x) - 144*sqrt(sqrt(2) - 1)*log(sqrt (sqrt(2) - 1) + x) - 64*x**3 + 768*x)/192