Integrand size = 15, antiderivative size = 249 \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}+\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}-\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}} \] Output:
-1/4*x^(3/2)/c/(c*x^4+a)-3/32*arctan(-1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8) )*2^(1/2)/(-a)^(5/8)/c^(11/8)-3/32*arctan(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^( 1/8))*2^(1/2)/(-a)^(5/8)/c^(11/8)+3/16*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/ (-a)^(5/8)/c^(11/8)-3/16*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(5/8)/c^ (11/8)+3/32*arctanh(2^(1/2)*(-a)^(1/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4) *x))*2^(1/2)/(-a)^(5/8)/c^(11/8)
Time = 1.32 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.11 \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 c^{3/8} x^{3/2}}{a+c x^4}+\frac {3 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{a^{5/8}}-\frac {3 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{a^{5/8}}+\frac {3 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{a^{5/8}}-\frac {3 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{a^{5/8}}}{32 c^{11/8}} \] Input:
Integrate[x^(9/2)/(a + c*x^4)^2,x]
Output:
((-8*c^(3/8)*x^(3/2))/(a + c*x^4) + (3*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])])/a^(5/8) - (3 *Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^( 1/8)*c^(1/8)*Sqrt[x])])/a^(5/8) + (3*Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 + S qrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)])/a^(5/8) - (3*Sqrt [2 + Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/a^(5/8))/(32*c^(11/8))
Time = 1.02 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.43, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {817, 851, 829, 826, 827, 218, 221, 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.
| \(\displaystyle \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {x}}{c x^4+a}dx}{8 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {3 \int \frac {x}{c x^4+a}d\sqrt {x}}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 829 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\int \frac {x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}\right )}{4 c}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}\) |
Input:
Int[x^(9/2)/(a + c*x^4)^2,x]
Output:
-1/4*x^(3/2)/(c*(a + c*x^4)) + (3*(-1/2*(-1/2*ArcTan[(c^(1/8)*Sqrt[x])/(-a )^(1/8)]/((-a)^(1/8)*c^(3/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*( -a)^(1/8)*c^(3/8)))/Sqrt[-a] - ((-(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(- a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^(1/8))) + ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt [x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^(1/8)))/(2*c^(1/4)) - (-1/2*Log[(-a )^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(Sqrt[2]*(-a)^(1 /8)*c^(1/8)) + Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/ 4)*x]/(2*Sqrt[2]*(-a)^(1/8)*c^(1/8)))/(2*c^(1/4)))/(2*Sqrt[-a])))/(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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/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[(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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[x^m/(r + s*x^ (n/2)), x], x] + Simp[r/(2*a) Int[x^m/(r - s*x^(n/2)), x], x]] /; FreeQ[{ a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && 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 = 0.51 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.19
| method | result | size |
| derivativedivides | \(-\frac {x^{\frac {3}{2}}}{4 c \left (c \,x^{4}+a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{32 c^{2}}\) | \(47\) |
| default | \(-\frac {x^{\frac {3}{2}}}{4 c \left (c \,x^{4}+a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{32 c^{2}}\) | \(47\) |
Input:
int(x^(9/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*x^(3/2)/c/(c*x^4+a)+3/32/c^2*sum(1/_R^5*ln(x^(1/2)-_R),_R=RootOf(_Z^8 *c+a))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.70 \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {3 \, \sqrt {2} {\left (\left (i - 1\right ) \, c^{2} x^{4} + \left (i - 1\right ) \, a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 3 \, \sqrt {2} {\left (-\left (i + 1\right ) \, c^{2} x^{4} - \left (i + 1\right ) \, a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 3 \, \sqrt {2} {\left (\left (i + 1\right ) \, c^{2} x^{4} + \left (i + 1\right ) \, a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 3 \, \sqrt {2} {\left (-\left (i - 1\right ) \, c^{2} x^{4} - \left (i - 1\right ) \, a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 6 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 6 \, {\left (-i \, c^{2} x^{4} - i \, a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (i \, a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 6 \, {\left (i \, c^{2} x^{4} + i \, a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (-i \, a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - 6 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (-a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 16 \, x^{\frac {3}{2}}}{64 \, {\left (c^{2} x^{4} + a c\right )}} \] Input:
integrate(x^(9/2)/(c*x^4+a)^2,x, algorithm="fricas")
Output:
-1/64*(3*sqrt(2)*((I - 1)*c^2*x^4 + (I - 1)*a*c)*(-1/(a^5*c^11))^(1/8)*log ((1/2*I + 1/2)*sqrt(2)*a^2*c^4*(-1/(a^5*c^11))^(3/8) + sqrt(x)) + 3*sqrt(2 )*(-(I + 1)*c^2*x^4 - (I + 1)*a*c)*(-1/(a^5*c^11))^(1/8)*log(-(1/2*I - 1/2 )*sqrt(2)*a^2*c^4*(-1/(a^5*c^11))^(3/8) + sqrt(x)) + 3*sqrt(2)*((I + 1)*c^ 2*x^4 + (I + 1)*a*c)*(-1/(a^5*c^11))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^2*c ^4*(-1/(a^5*c^11))^(3/8) + sqrt(x)) + 3*sqrt(2)*(-(I - 1)*c^2*x^4 - (I - 1 )*a*c)*(-1/(a^5*c^11))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a^2*c^4*(-1/(a^5*c ^11))^(3/8) + sqrt(x)) + 6*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(a^2*c ^4*(-1/(a^5*c^11))^(3/8) + sqrt(x)) + 6*(-I*c^2*x^4 - I*a*c)*(-1/(a^5*c^11 ))^(1/8)*log(I*a^2*c^4*(-1/(a^5*c^11))^(3/8) + sqrt(x)) + 6*(I*c^2*x^4 + I *a*c)*(-1/(a^5*c^11))^(1/8)*log(-I*a^2*c^4*(-1/(a^5*c^11))^(3/8) + sqrt(x) ) - 6*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(-a^2*c^4*(-1/(a^5*c^11))^( 3/8) + sqrt(x)) + 16*x^(3/2))/(c^2*x^4 + a*c)
Timed out. \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**(9/2)/(c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {9}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate(x^(9/2)/(c*x^4+a)^2,x, algorithm="maxima")
Output:
-1/4*x^(3/2)/(c^2*x^4 + a*c) + 3*integrate(1/8*sqrt(x)/(c^2*x^4 + a*c), x)
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (168) = 336\).
Time = 0.25 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.95 \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{\frac {3}{2}}}{4 \, {\left (c x^{4} + a\right )} c} - \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {2 \, \sqrt {2} + 4}} + \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {3 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a c \sqrt {-2 \, \sqrt {2} + 4}} \] Input:
integrate(x^(9/2)/(c*x^4+a)^2,x, algorithm="giac")
Output:
-1/4*x^(3/2)/((c*x^4 + a)*c) - 3/16*(a/c)^(3/8)*arctan((sqrt(-sqrt(2) + 2) *(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(2*sqr t(2) + 4)) - 3/16*(a/c)^(3/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2* sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(2*sqrt(2) + 4)) + 3/16 *(a/c)^(3/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqr t(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(-2*sqrt(2) + 4)) + 3/16*(a/c)^(3/8)*arct an(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^ (1/8)))/(a*c*sqrt(-2*sqrt(2) + 4)) + 3/32*(a/c)^(3/8)*log(sqrt(x)*sqrt(sqr t(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(2*sqrt(2) + 4)) - 3/32* (a/c)^(3/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/ (a*c*sqrt(2*sqrt(2) + 4)) - 3/32*(a/c)^(3/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2 )*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(-2*sqrt(2) + 4)) + 3/32*(a/c)^( 3/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*s qrt(-2*sqrt(2) + 4))
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.54 \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx=\frac {3\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{5/8}\,c^{11/8}}-\frac {x^{3/2}}{4\,c\,\left (c\,x^4+a\right )}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{5/8}\,c^{11/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {3}{32}+\frac {3}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,c^{11/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {3}{32}-\frac {3}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,c^{11/8}} \] Input:
int(x^(9/2)/(a + c*x^4)^2,x)
Output:
(3*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(16*(-a)^(5/8)*c^(11/8)) - x^(3/2)/ (4*c*(a + c*x^4)) + (atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*3i)/(16*(-a)^(5 /8)*c^(11/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^ (1/8))*(3/32 - 3i/32))/((-a)^(5/8)*c^(11/8)) - (2^(1/2)*atan((2^(1/2)*c^(1 /8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(3/32 + 3i/32))/((-a)^(5/8)*c^(11/8) )
Time = 0.28 (sec) , antiderivative size = 1553, normalized size of antiderivative = 6.24 \[ \int \frac {x^{9/2}}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(x^(9/2)/(c*x^4+a)^2,x)
Output:
(6*c**(5/8)*a**(3/8)*sqrt(sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqr t( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*a + 6*c**(5/8)*a**(3/8)*sqrt(sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**( 1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sq rt(2) + 2)))*c*x**4 - 6*c**(5/8)*a**(3/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8) *a**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sq rt(sqrt(2) + 2)))*a - 6*c**(5/8)*a**(3/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8) *a**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sq rt(sqrt(2) + 2)))*c*x**4 - 6*c**(5/8)*a**(3/8)*sqrt(sqrt(2) + 2)*sqrt(2)*a tan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8 )*a**(1/8)*sqrt(sqrt(2) + 2)))*a - 6*c**(5/8)*a**(3/8)*sqrt(sqrt(2) + 2)*s qrt(2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/ (c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*c*x**4 + 6*c**(5/8)*a**(3/8)*sqrt(s qrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1 /4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*a + 6*c**(5/8)*a**(3/8)*sqrt(s qrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1 /4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*c*x**4 - 6*c**(5/8)*a**(3/8)*s qrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) - 2* sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*a - 6*c**(5/8) *a**(3/8)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(sqr...