Integrand size = 13, antiderivative size = 201 \[ \int \frac {x^3}{a^5+x^5} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a}-\frac {\log (a+x)}{5 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a} \] Output:
-1/5*ln(a+x)/a+1/20*ln(a^2+x^2-1/2*a*x*(-5^(1/2)+1))*(-5^(1/2)+1)/a+1/20*l n(a^2+x^2-1/2*a*x*(5^(1/2)+1))*(5^(1/2)+1)/a-1/10*arctan(1/20*(-4*x+a*(5^( 1/2)+1))*(50+10*5^(1/2))^(1/2)/a)*(10-2*5^(1/2))^(1/2)/a-1/10*arctan((-4*x +a*(-5^(1/2)+1))/a/(10+2*5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)/a
Time = 0.02 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (-1+\sqrt {5}\right ) a+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )+2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {-\left (\left (1+\sqrt {5}\right ) a\right )+4 x}{\sqrt {10-2 \sqrt {5}} a}\right )-4 \log (a+x)+\log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )-\sqrt {5} \log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )+\log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )+\sqrt {5} \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a} \] Input:
Integrate[x^3/(a^5 + x^5),x]
Output:
(2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[((-1 + Sqrt[5])*a + 4*x)/(Sqrt[2*(5 + Sqrt [5])]*a)] + 2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(-((1 + Sqrt[5])*a) + 4*x)/(Sqrt [10 - 2*Sqrt[5]]*a)] - 4*Log[a + x] + Log[a^2 + ((-1 + Sqrt[5])*a*x)/2 + x ^2] - Sqrt[5]*Log[a^2 + ((-1 + Sqrt[5])*a*x)/2 + x^2] + Log[a^2 - ((1 + Sq rt[5])*a*x)/2 + x^2] + Sqrt[5]*Log[a^2 - ((1 + Sqrt[5])*a*x)/2 + x^2])/(20 *a)
Time = 0.37 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {822, 16, 27, 1142, 25, 1083, 217, 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.
| \(\displaystyle \int \frac {x^3}{a^5+x^5} \, dx\) |
| \(\Big \downarrow \) 822 |
| \(\displaystyle \frac {2 \int \frac {\left (1+\sqrt {5}\right ) a+\left (1-\sqrt {5}\right ) x}{2 \left (2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2\right )}dx}{5 a}+\frac {2 \int \frac {\left (1-\sqrt {5}\right ) a+\left (1+\sqrt {5}\right ) x}{2 \left (2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2\right )}dx}{5 a}-\frac {\int \frac {1}{a+x}dx}{5 a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \int \frac {\left (1+\sqrt {5}\right ) a+\left (1-\sqrt {5}\right ) x}{2 \left (2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2\right )}dx}{5 a}+\frac {2 \int \frac {\left (1-\sqrt {5}\right ) a+\left (1+\sqrt {5}\right ) x}{2 \left (2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2\right )}dx}{5 a}-\frac {\log (a+x)}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (1+\sqrt {5}\right ) a+\left (1-\sqrt {5}\right ) x}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx}{5 a}+\frac {\int \frac {\left (1-\sqrt {5}\right ) a+\left (1+\sqrt {5}\right ) x}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx}{5 a}-\frac {\log (a+x)}{5 a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) a \int \frac {1}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int -\frac {\left (1-\sqrt {5}\right ) a-4 x}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx}{5 a}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) a \int \frac {1}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int -\frac {\left (1+\sqrt {5}\right ) a-4 x}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx}{5 a}-\frac {\log (a+x)}{5 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (5+\sqrt {5}\right ) a \int \frac {1}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) a-4 x}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx}{5 a}+\frac {\frac {1}{2} \left (5-\sqrt {5}\right ) a \int \frac {1}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) a-4 x}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx}{5 a}-\frac {\log (a+x)}{5 a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) a-4 x}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx-\left (5+\sqrt {5}\right ) a \int \frac {1}{-2 \left (5+\sqrt {5}\right ) a^2-\left (4 x-\left (1-\sqrt {5}\right ) a\right )^2}d\left (4 x-\left (1-\sqrt {5}\right ) a\right )}{5 a}+\frac {-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) a-4 x}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx-\left (5-\sqrt {5}\right ) a \int \frac {1}{-2 \left (5-\sqrt {5}\right ) a^2-\left (4 x-\left (1+\sqrt {5}\right ) a\right )^2}d\left (4 x-\left (1+\sqrt {5}\right ) a\right )}{5 a}-\frac {\log (a+x)}{5 a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x-\left (1-\sqrt {5}\right ) a}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {\left (1-\sqrt {5}\right ) a-4 x}{2 a^2-\left (1-\sqrt {5}\right ) x a+2 x^2}dx}{5 a}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\left (1+\sqrt {5}\right ) a}{\sqrt {2 \left (5-\sqrt {5}\right )} a}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {\left (1+\sqrt {5}\right ) a-4 x}{2 a^2-\left (1+\sqrt {5}\right ) x a+2 x^2}dx}{5 a}-\frac {\log (a+x)}{5 a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) \log \left (2 a^2-\left (1-\sqrt {5}\right ) a x+2 x^2\right )+\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x-\left (1-\sqrt {5}\right ) a}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}+\frac {\frac {1}{4} \left (1+\sqrt {5}\right ) \log \left (2 a^2-\left (1+\sqrt {5}\right ) a x+2 x^2\right )+\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\left (1+\sqrt {5}\right ) a}{\sqrt {2 \left (5-\sqrt {5}\right )} a}\right )}{5 a}-\frac {\log (a+x)}{5 a}\) |
Input:
Int[x^3/(a^5 + x^5),x]
Output:
-1/5*Log[a + x]/a + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(-((1 - Sqrt[5])*a) + 4* x)/(Sqrt[2*(5 + Sqrt[5])]*a)] + ((1 - Sqrt[5])*Log[2*a^2 - (1 - Sqrt[5])*a *x + 2*x^2])/4)/(5*a) + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[(-((1 + Sqrt[5])*a) + 4*x)/(Sqrt[2*(5 - Sqrt[5])]*a)] + ((1 + Sqrt[5])*Log[2*a^2 - (1 + Sqrt[5 ])*a*x + 2*x^2])/4)/(5*a)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; -(-r)^(m + 1)/(a*n*s^m) Int[1/(r + s*x), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 1)/2}], x]] /; FreeQ[{a, b} , x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{4}-a^{3} \textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3} a^{4}-\textit {\_R}^{2} a^{3}+\textit {\_R} \,a^{2}-a +x \right )\right )}{5}-\frac {\ln \left (a +x \right )}{5 a}\) | \(73\) |
| default | \(-\frac {\ln \left (a +x \right )}{5 a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}+3 \textit {\_R}^{2} a -2 \textit {\_R} \,a^{2}+a^{3}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 \textit {\_R} \,a^{2}-a^{3}}}{5 a}\) | \(97\) |
Input:
int(x^3/(a^5+x^5),x,method=_RETURNVERBOSE)
Output:
1/5*sum(_R*ln(_R^3*a^4-_R^2*a^3+_R*a^2-a+x),_R=RootOf(_Z^4*a^4-_Z^3*a^3+_Z ^2*a^2-_Z*a+1))-1/5*ln(a+x)/a
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 17865, normalized size of antiderivative = 88.88 \[ \int \frac {x^3}{a^5+x^5} \, dx=\text {Too large to display} \] Input:
integrate(x^3/(a^5+x^5),x, algorithm="fricas")
Output:
Too large to include
Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.19 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {- \frac {\log {\left (a + x \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log {\left (625 t^{4} a + x \right )} \right )\right )}}{a} \] Input:
integrate(x**3/(a**5+x**5),x)
Output:
(-log(a + x)/5 + RootSum(625*_t**4 - 125*_t**3 + 25*_t**2 - 5*_t + 1, Lamb da(_t, _t*log(625*_t**4*a + x))))/a
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {-2 \, \sqrt {5} + 10}} + \frac {{\left (\sqrt {5} + 3\right )} \log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} + 1\right )}} + \frac {{\left (\sqrt {5} - 3\right )} \log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} - 1\right )}} - \frac {\log \left (a + x\right )}{5 \, a} \] Input:
integrate(x^3/(a^5+x^5),x, algorithm="maxima")
Output:
1/5*sqrt(5)*(sqrt(5) + 1)*arctan((a*(sqrt(5) - 1) + 4*x)/(a*sqrt(2*sqrt(5) + 10)))/(a*sqrt(2*sqrt(5) + 10)) + 1/5*sqrt(5)*(sqrt(5) - 1)*arctan(-(a*( sqrt(5) + 1) - 4*x)/(a*sqrt(-2*sqrt(5) + 10)))/(a*sqrt(-2*sqrt(5) + 10)) + 1/10*(sqrt(5) + 3)*log(-a*x*(sqrt(5) + 1) + 2*a^2 + 2*x^2)/(a*(sqrt(5) + 1)) + 1/10*(sqrt(5) - 3)*log(a*x*(sqrt(5) - 1) + 2*a^2 + 2*x^2)/(a*(sqrt(5 ) - 1)) - 1/5*log(a + x)/a
Time = 0.13 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{10 \, a} + \frac {\sqrt {-2 \, \sqrt {5} + 10} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{10 \, a} + \frac {\sqrt {5} \log \left (a^{2} - \frac {1}{2} \, {\left (\sqrt {5} a + a\right )} x + x^{2}\right )}{20 \, a} - \frac {\sqrt {5} \log \left (a^{2} + \frac {1}{2} \, {\left (\sqrt {5} a - a\right )} x + x^{2}\right )}{20 \, a} + \frac {\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a} - \frac {\log \left ({\left | a + x \right |}\right )}{5 \, a} \] Input:
integrate(x^3/(a^5+x^5),x, algorithm="giac")
Output:
1/10*sqrt(2*sqrt(5) + 10)*arctan((a*(sqrt(5) - 1) + 4*x)/(a*sqrt(2*sqrt(5) + 10)))/a + 1/10*sqrt(-2*sqrt(5) + 10)*arctan(-(a*(sqrt(5) + 1) - 4*x)/(a *sqrt(-2*sqrt(5) + 10)))/a + 1/20*sqrt(5)*log(a^2 - 1/2*(sqrt(5)*a + a)*x + x^2)/a - 1/20*sqrt(5)*log(a^2 + 1/2*(sqrt(5)*a - a)*x + x^2)/a + 1/20*lo g(abs(a^4 - a^3*x + a^2*x^2 - a*x^3 + x^4))/a - 1/5*log(abs(a + x))/a
Time = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}-\frac {\ln \left (5\,a^{10}+\frac {5\,x\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )\,a^9}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a}-\frac {\ln \left (a+x\right )}{5\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a} \] Input:
int(x^3/(a^5 + x^5),x)
Output:
(log(5*a^10 - (5*a^9*x*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1))/4)*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1))/(20*a) - (log(5*a^10 + (5*a^9*x*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) - 1))/4)*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) - 1 ))/(20*a) - log(a + x)/(5*a) + (log(5*a^10 - (5*a^9*x*(5^(1/2) - (2*5^(1/2 ) - 10)^(1/2) + 1))/4)*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) + 1))/(20*a) + (l og(5*a^10 - (5*a^9*x*((- 2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1))/4)*((- 2*5^ (1/2) - 10)^(1/2) - 5^(1/2) + 1))/(20*a)
\[ \int \frac {x^3}{a^5+x^5} \, dx=\int \frac {x^{3}}{a^{5}+x^{5}}d x \] Input:
int(x^3/(a^5+x^5),x)
Output:
int(x**3/(a**5 + x**5),x)