Integrand size = 13, antiderivative size = 144 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {x^3}{3 c}+\frac {a^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4}}-\frac {a^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4}}+\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} c^{7/4}} \] Output:
1/3*x^3/c-1/4*a^(3/4)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/c^(7/4) -1/4*a^(3/4)*arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/c^(7/4)+1/4*a^(3/ 4)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))*2^(1/2)/c^(7/4 )
Time = 0.03 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.22 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {8 c^{3/4} x^3+6 \sqrt {2} a^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-6 \sqrt {2} a^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-3 \sqrt {2} a^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+3 \sqrt {2} a^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{24 c^{7/4}} \] Input:
Integrate[x^6/(a + c*x^4),x]
Output:
(8*c^(3/4)*x^3 + 6*Sqrt[2]*a^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 6*Sqrt[2]*a^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*a ^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 3*Sqrt[2]* a^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(24*c^(7/4 ))
Time = 0.69 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.51, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {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.
| \(\displaystyle \int \frac {x^6}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \int \frac {x^2}{c x^4+a}dx}{c}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}}{2 \sqrt {c}}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^3}{3 c}-\frac {a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\) |
Input:
Int[x^6/(a + c*x^4),x]
Output:
x^3/(3*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[c]) - (-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[c])))/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[(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]]))
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 = 0.69 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.26
| method | result | size |
| risch | \(\frac {x^{3}}{3 c}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{4 c^{2}}\) | \(37\) |
| default | \(\frac {x^{3}}{3 c}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c^{2} \left (\frac {a}{c}\right )^{\frac {1}{4}}}\) | \(112\) |
Input:
int(x^6/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/3*x^3/c-1/4/c^2*a*sum(1/_R*ln(x-_R),_R=RootOf(_Z^4*c+a))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {4 \, x^{3} - 3 \, c \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {1}{4}} \log \left (c^{5} \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {3}{4}} + a^{2} x\right ) + 3 i \, c \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {1}{4}} \log \left (i \, c^{5} \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {3}{4}} + a^{2} x\right ) - 3 i \, c \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {1}{4}} \log \left (-i \, c^{5} \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {3}{4}} + a^{2} x\right ) + 3 \, c \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {1}{4}} \log \left (-c^{5} \left (-\frac {a^{3}}{c^{7}}\right )^{\frac {3}{4}} + a^{2} x\right )}{12 \, c} \] Input:
integrate(x^6/(c*x^4+a),x, algorithm="fricas")
Output:
1/12*(4*x^3 - 3*c*(-a^3/c^7)^(1/4)*log(c^5*(-a^3/c^7)^(3/4) + a^2*x) + 3*I *c*(-a^3/c^7)^(1/4)*log(I*c^5*(-a^3/c^7)^(3/4) + a^2*x) - 3*I*c*(-a^3/c^7) ^(1/4)*log(-I*c^5*(-a^3/c^7)^(3/4) + a^2*x) + 3*c*(-a^3/c^7)^(1/4)*log(-c^ 5*(-a^3/c^7)^(3/4) + a^2*x))/c
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.24 \[ \int \frac {x^6}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} c^{7} + a^{3}, \left ( t \mapsto t \log {\left (- \frac {64 t^{3} c^{5}}{a^{2}} + x \right )} \right )\right )} + \frac {x^{3}}{3 c} \] Input:
integrate(x**6/(c*x**4+a),x)
Output:
RootSum(256*_t**4*c**7 + a**3, Lambda(_t, _t*log(-64*_t**3*c**5/a**2 + x)) ) + x**3/(3*c)
Time = 0.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.27 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {x^{3}}{3 \, c} - \frac {a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{8 \, c} \] Input:
integrate(x^6/(c*x^4+a),x, algorithm="maxima")
Output:
1/3*x^3/c - 1/8*a*(2*sqrt(2)*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(c)) + 2*s qrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqr t(a)*sqrt(c)))/(sqrt(sqrt(a)*sqrt(c))*sqrt(c)) - sqrt(2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(1/4)*c^(3/4)) + sqrt(2)*log(sqrt (c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(1/4)*c^(3/4)))/c
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.22 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {x^{3}}{3 \, c} - \frac {\sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, c^{4}} - \frac {\sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, c^{4}} + \frac {\sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, c^{4}} - \frac {\sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, c^{4}} \] Input:
integrate(x^6/(c*x^4+a),x, algorithm="giac")
Output:
1/3*x^3/c - 1/4*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a /c)^(1/4))/(a/c)^(1/4))/c^4 - 1/4*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2) *(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/c^4 + 1/8*sqrt(2)*(a*c^3)^(3/4)* log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/c^4 - 1/8*sqrt(2)*(a*c^3)^(3/ 4)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/c^4
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.35 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {x^3}{3\,c}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{2\,c^{7/4}}-\frac {{\left (-a\right )}^{3/4}\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{2\,c^{7/4}} \] Input:
int(x^6/(a + c*x^4),x)
Output:
x^3/(3*c) + ((-a)^(3/4)*atan((c^(1/4)*x)/(-a)^(1/4)))/(2*c^(7/4)) - ((-a)^ (3/4)*atanh((c^(1/4)*x)/(-a)^(1/4)))/(2*c^(7/4))
Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.02 \[ \int \frac {x^6}{a+c x^4} \, dx=\frac {6 c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-6 c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-3 c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right )+3 c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right )+8 c \,x^{3}}{24 c^{2}} \] Input:
int(x^6/(c*x^4+a),x)
Output:
(6*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x )/(c**(1/4)*a**(1/4)*sqrt(2))) - 6*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4 )*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2))) - 3*c**(1/4 )*a**(3/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)* x**2) + 3*c**(1/4)*a**(3/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt (a) + sqrt(c)*x**2) + 8*c*x**3)/(24*c**2)