Integrand size = 13, antiderivative size = 157 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {x}{c^2}+\frac {a x}{4 c^2 \left (a+c x^4\right )}+\frac {5 \sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{9/4}}-\frac {5 \sqrt [4]{a} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{9/4}}-\frac {5 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{8 \sqrt {2} c^{9/4}} \] Output:
x/c^2+1/4*a*x/c^2/(c*x^4+a)-5/16*a^(1/4)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/ 4))*2^(1/2)/c^(9/4)-5/16*a^(1/4)*arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/ 2)/c^(9/4)-5/16*a^(1/4)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2) *x^2))*2^(1/2)/c^(9/4)
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.22 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {32 \sqrt [4]{c} x+\frac {8 a \sqrt [4]{c} x}{a+c x^4}+10 \sqrt {2} \sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-10 \sqrt {2} \sqrt [4]{a} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+5 \sqrt {2} \sqrt [4]{a} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-5 \sqrt {2} \sqrt [4]{a} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{32 c^{9/4}} \] Input:
Integrate[x^8/(a + c*x^4)^2,x]
Output:
(32*c^(1/4)*x + (8*a*c^(1/4)*x)/(a + c*x^4) + 10*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 10*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt[2]*c^ (1/4)*x)/a^(1/4)] + 5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4 )*x + Sqrt[c]*x^2] - 5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/ 4)*x + Sqrt[c]*x^2])/(32*c^(9/4))
Time = 0.68 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.52, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {817, 843, 755, 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.
| \(\displaystyle \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {5 \int \frac {x^4}{c x^4+a}dx}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \int \frac {1}{c x^4+a}dx}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {x}{c}-\frac {a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {a}}\right )}{c}\right )}{4 c}-\frac {x^5}{4 c \left (a+c x^4\right )}\) |
Input:
Int[x^8/(a + c*x^4)^2,x]
Output:
-1/4*x^5/(c*(a + c*x^4)) + (5*(x/c - (a*((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*x) /a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1 /4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]* a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(Sqrt[2]*a^(1/4)*c^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(2*Sqrt[2]*a^(1/4)*c^(1/4)))/(2* Sqrt[a])))/c))/(4*c)
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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]]))
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]
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]
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]
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_)^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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.47 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.32
| method | result | size |
| risch | \(\frac {x}{c^{2}}+\frac {a x}{4 c^{2} \left (c \,x^{4}+a \right )}-\frac {5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16 c^{3}}\) | \(50\) |
| default | \(\frac {x}{c^{2}}-\frac {a \left (-\frac {x}{4 \left (c \,x^{4}+a \right )}+\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{c^{2}}\) | \(127\) |
Input:
int(x^8/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
x/c^2+1/4*a*x/c^2/(c*x^4+a)-5/16/c^3*a*sum(1/_R^3*ln(x-_R),_R=RootOf(_Z^4* c+a))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.25 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {16 \, c x^{5} - 5 \, {\left (c^{3} x^{4} + a c^{2}\right )} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} \log \left (5 \, c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + 5 \, x\right ) - 5 \, {\left (i \, c^{3} x^{4} + i \, a c^{2}\right )} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} \log \left (5 i \, c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + 5 \, x\right ) - 5 \, {\left (-i \, c^{3} x^{4} - i \, a c^{2}\right )} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} \log \left (-5 i \, c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + 5 \, x\right ) + 5 \, {\left (c^{3} x^{4} + a c^{2}\right )} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} \log \left (-5 \, c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + 5 \, x\right ) + 20 \, a x}{16 \, {\left (c^{3} x^{4} + a c^{2}\right )}} \] Input:
integrate(x^8/(c*x^4+a)^2,x, algorithm="fricas")
Output:
1/16*(16*c*x^5 - 5*(c^3*x^4 + a*c^2)*(-a/c^9)^(1/4)*log(5*c^2*(-a/c^9)^(1/ 4) + 5*x) - 5*(I*c^3*x^4 + I*a*c^2)*(-a/c^9)^(1/4)*log(5*I*c^2*(-a/c^9)^(1 /4) + 5*x) - 5*(-I*c^3*x^4 - I*a*c^2)*(-a/c^9)^(1/4)*log(-5*I*c^2*(-a/c^9) ^(1/4) + 5*x) + 5*(c^3*x^4 + a*c^2)*(-a/c^9)^(1/4)*log(-5*c^2*(-a/c^9)^(1/ 4) + 5*x) + 20*a*x)/(c^3*x^4 + a*c^2)
Time = 0.17 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.31 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {a x}{4 a c^{2} + 4 c^{3} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} c^{9} + 625 a, \left ( t \mapsto t \log {\left (- \frac {16 t c^{2}}{5} + x \right )} \right )\right )} + \frac {x}{c^{2}} \] Input:
integrate(x**8/(c*x**4+a)**2,x)
Output:
a*x/(4*a*c**2 + 4*c**3*x**4) + RootSum(65536*_t**4*c**9 + 625*a, Lambda(_t , _t*log(-16*_t*c**2/5 + x))) + x/c**2
Time = 0.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.26 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {a x}{4 \, {\left (c^{3} x^{4} + a c^{2}\right )}} - \frac {5 \, {\left (\frac {2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{c^{\frac {1}{4}}}\right )}}{32 \, c^{2}} + \frac {x}{c^{2}} \] Input:
integrate(x^8/(c*x^4+a)^2,x, algorithm="maxima")
Output:
1/4*a*x/(c^3*x^4 + a*c^2) - 5/32*(2*sqrt(2)*sqrt(a)*arctan(1/2*sqrt(2)*(2* sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/sqrt(sqrt(a)*s qrt(c)) + 2*sqrt(2)*sqrt(a)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1 /4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/sqrt(sqrt(a)*sqrt(c)) + sqrt(2)*a^(1/4 )*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/c^(1/4) - sqrt(2) *a^(1/4)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/c^(1/4))/c ^2 + x/c^2
Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.20 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {a x}{4 \, {\left (c x^{4} + a\right )} c^{2}} + \frac {x}{c^{2}} - \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, c^{3}} - \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, c^{3}} - \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, c^{3}} + \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, c^{3}} \] Input:
integrate(x^8/(c*x^4+a)^2,x, algorithm="giac")
Output:
1/4*a*x/((c*x^4 + a)*c^2) + x/c^2 - 5/16*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2* sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/c^3 - 5/16*sqrt(2)*(a*c^3 )^(1/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/c^3 - 5/32*sqrt(2)*(a*c^3)^(1/4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/c^ 3 + 5/32*sqrt(2)*(a*c^3)^(1/4)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c) )/c^3
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.45 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {x}{c^2}-\frac {5\,{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{8\,c^{9/4}}+\frac {a\,x}{4\,\left (c^3\,x^4+a\,c^2\right )}+\frac {{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}}\right )\,5{}\mathrm {i}}{8\,c^{9/4}} \] Input:
int(x^8/(a + c*x^4)^2,x)
Output:
x/c^2 - (5*(-a)^(1/4)*atan((c^(1/4)*x)/(-a)^(1/4)))/(8*c^(9/4)) + ((-a)^(1 /4)*atan((c^(1/4)*x*1i)/(-a)^(1/4))*5i)/(8*c^(9/4)) + (a*x)/(4*(a*c^2 + c^ 3*x^4))
Time = 0.19 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.97 \[ \int \frac {x^8}{\left (a+c x^4\right )^2} \, dx=\frac {10 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+10 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{4}-10 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-10 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{4}+5 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right )+5 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) x^{4}-5 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right )-5 c^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) x^{4}+40 a c x +32 c^{2} x^{5}}{32 c^{3} \left (c \,x^{4}+a \right )} \] Input:
int(x^8/(c*x^4+a)^2,x)
Output:
(10*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)* x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a + 10*c**(3/4)*a**(1/4)*sqrt(2)*atan((c** (1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c*x**4 - 10*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c) *x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a - 10*c**(3/4)*a**(1/4)*sqrt(2)*atan((c* *(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c*x**4 + 5*c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a ) + sqrt(c)*x**2)*a + 5*c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4)*a**(1/4) *sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*c*x**4 - 5*c**(3/4)*a**(1/4)*sqrt(2)* log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a - 5*c**(3/4)*a **(1/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)* c*x**4 + 40*a*c*x + 32*c**2*x**5)/(32*c**3*(a + c*x**4))