Integrand size = 11, antiderivative size = 109 \[ \int \frac {1+x}{1+x^5} \, dx=-\frac {1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac {1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-(-1)^{4/5}-x\right )+\frac {1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left ((-1)^{2/5}+x\right )-\frac {1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (-(-1)^{3/5}+x\right ) \]
-1/5*(-1)^(1/5)*(1+(-1)^(1/5))*ln((-1)^(1/5)-x)+1/5*(-1)^(4/5)*(1-(-1)^(4/ 5))*ln(-(-1)^(4/5)-x)+1/5*(-1)^(2/5)*(1-(-1)^(2/5))*ln((-1)^(2/5)+x)-1/5*( -1)^(3/5)*(1+(-1)^(3/5))*ln(-(-1)^(3/5)+x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.47 \[ \int \frac {1+x}{1+x^5} \, dx=\text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{-1+2 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
Time = 0.44 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2019, 2492, 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.
\(\displaystyle \int \frac {x+1}{x^5+1} \, dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int \frac {1}{x^4-x^3+x^2-x+1}dx\) |
\(\Big \downarrow \) 2492 |
\(\displaystyle \int \left (\frac {-2 x+\sqrt {5}+1}{\sqrt {5} \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}-\frac {-2 x-\sqrt {5}+1}{\sqrt {5} \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{5} \sqrt {5-2 \sqrt {5}} \arctan \left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{5} \sqrt {5+2 \sqrt {5}} \arctan \left (\frac {-4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {\log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}{2 \sqrt {5}}-\frac {\log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}{2 \sqrt {5}}\) |
-1/5*(Sqrt[5 - 2*Sqrt[5]]*ArcTan[(1 - Sqrt[5] - 4*x)/Sqrt[2*(5 + Sqrt[5])] ]) - (Sqrt[5 + 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[2*(5 - Sqrt[5])] ])/5 + Log[2 - (1 - Sqrt[5])*x + 2*x^2]/(2*Sqrt[5]) - Log[2 - (1 + Sqrt[5] )*x + 2*x^2]/(2*Sqrt[5])
3.3.21.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.49 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} -1}\) | \(45\) |
default | \(\frac {\sqrt {5}\, \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{10}+\frac {2 \left (-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right )}{2}+5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\sqrt {5}\, \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{10}-\frac {2 \left (-\frac {\sqrt {5}\, \left (-\sqrt {5}-1\right )}{2}-\sqrt {5}-5\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(143\) |
meijerg | \(\frac {x \ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}-\frac {x \cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 x \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {x \cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 x \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}-\frac {x^{2} \ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {x^{2} \cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}+\frac {2 x^{2} \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}+\frac {x^{2} \cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {2 x^{2} \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}\) | \(318\) |
Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (73) = 146\).
Time = 0.99 (sec) , antiderivative size = 835, normalized size of antiderivative = 7.66 \[ \int \frac {1+x}{1+x^5} \, dx=\text {Too large to display} \]
-1/10*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))*log(3/8*(sqrt(5) + 5*sqrt(-2 /25*sqrt(5) - 1/5))^3 + 1/8*(3*sqrt(5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8) *(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2 + 3/8*((sqrt(5) + 5*sqrt(-2/25* sqrt(5) - 1/5))^2 - 12)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) + 11*x + 1 ) - 1/10*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))*log(-3/8*(sqrt(5) + 5*sqr t(-2/25*sqrt(5) - 1/5))^3 + (sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 + 11 *x - 9/2*sqrt(5) - 45/2*sqrt(-2/25*sqrt(5) - 1/5) - 14) + 1/10*(sqrt(5) + 5*sqrt(-3/100*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 1/50*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) - 3/1 00*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2))*log(-1/8*(3*sqrt(5) + 15*sq rt(-2/25*sqrt(5) - 1/5) + 8)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - ( sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 3/8*((sqrt(5) + 5*sqrt(-2/25*sq rt(5) - 1/5))^2 - 12)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) + 5/4*sqrt(- 3/100*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 1/50*(sqrt(5) + 5*sqrt(- 2/25*sqrt(5) - 1/5))*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) - 3/100*(sqrt (5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2)*((3*sqrt(5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) + 8*sqrt(5) + 40*sqrt( -2/25*sqrt(5) - 1/5) + 36) + 22*x + 9/2*sqrt(5) + 45/2*sqrt(-2/25*sqrt(5) - 1/5) + 2) + 1/10*(sqrt(5) - 5*sqrt(-3/100*(sqrt(5) + 5*sqrt(-2/25*sqrt(5 ) - 1/5))^2 - 1/50*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))*(sqrt(5) - 5...
Leaf count of result is larger than twice the leaf count of optimal. 1287 vs. \(2 (109) = 218\).
Time = 0.58 (sec) , antiderivative size = 1287, normalized size of antiderivative = 11.81 \[ \int \frac {1+x}{1+x^5} \, dx=\text {Too large to display} \]
sqrt(5)*log(x**2 + x*(-48/11 - 21*sqrt(5)/11 + 4*sqrt(10)*sqrt(sqrt(5) + 3 )/11 + 45*sqrt(2)*sqrt(sqrt(5) + 3)/22) - 1381*sqrt(10)*sqrt(sqrt(5) + 3)/ 484 - 3045*sqrt(2)*sqrt(sqrt(5) + 3)/484 + 2213*sqrt(5)/242 + 5217/242)/10 - sqrt(5)*log(x**2 + x*(-48/11 - 45*sqrt(2)*sqrt(3 - sqrt(5))/22 + 4*sqrt (10)*sqrt(3 - sqrt(5))/11 + 21*sqrt(5)/11) - 2213*sqrt(5)/242 - 1381*sqrt( 10)*sqrt(3 - sqrt(5))/484 + 3045*sqrt(2)*sqrt(3 - sqrt(5))/484 + 5217/242) /10 + 2*sqrt(-sqrt(10)*sqrt(3 - sqrt(5))/50 + 3/20)*atan(44*x/(-8*sqrt(5)* sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 3*sqrt(10)*sqrt(3 - sqrt(5))*sq rt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 18*sqrt(-2*sqrt(10)*sqrt(3 - sqrt (5)) + 15)) - 96/(-8*sqrt(5)*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 3* sqrt(10)*sqrt(3 - sqrt(5))*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 18*s qrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15)) - 45*sqrt(2)*sqrt(3 - sqrt(5))/(- 8*sqrt(5)*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 3*sqrt(10)*sqrt(3 - s qrt(5))*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 18*sqrt(-2*sqrt(10)*sqr t(3 - sqrt(5)) + 15)) + 8*sqrt(10)*sqrt(3 - sqrt(5))/(-8*sqrt(5)*sqrt(-2*s qrt(10)*sqrt(3 - sqrt(5)) + 15) + 3*sqrt(10)*sqrt(3 - sqrt(5))*sqrt(-2*sqr t(10)*sqrt(3 - sqrt(5)) + 15) + 18*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15 )) + 42*sqrt(5)/(-8*sqrt(5)*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 3*s qrt(10)*sqrt(3 - sqrt(5))*sqrt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15) + 18*sq rt(-2*sqrt(10)*sqrt(3 - sqrt(5)) + 15))) + 2*sqrt(-sqrt(10)*sqrt(sqrt(5...
\[ \int \frac {1+x}{1+x^5} \, dx=\int { \frac {x + 1}{x^{5} + 1} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \frac {1+x}{1+x^5} \, dx=\frac {1}{5} \, \sqrt {-2 \, \sqrt {5} + 5} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{5} \, \sqrt {2 \, \sqrt {5} + 5} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {1}{10} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{10} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) \]
1/5*sqrt(-2*sqrt(5) + 5)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) + 1/5*sqrt(2*sqrt(5) + 5)*arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10) ) - 1/10*sqrt(5)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/10*sqrt(5)*log(x^2 + 1/2*x*(sqrt(5) - 1) + 1)
Time = 9.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {1+x}{1+x^5} \, dx=\sum _{k=1}^4\ln \left (\mathrm {root}\left (z^4-\frac {z}{25}+\frac {1}{125},z,k\right )\,\left (-4\,x+\mathrm {root}\left (z^4-\frac {z}{25}+\frac {1}{125},z,k\right )\,\left (25\,\mathrm {root}\left (z^4-\frac {z}{25}+\frac {1}{125},z,k\right )+15\,x-15\right )+1\right )\right )\,\mathrm {root}\left (z^4-\frac {z}{25}+\frac {1}{125},z,k\right ) \]