Integrand size = 13, antiderivative size = 182 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=-\frac {1}{a^3 x}-\frac {c x^3}{8 a^2 \left (a+c x^4\right )^2}-\frac {13 c x^3}{32 a^3 \left (a+c x^4\right )}+\frac {45 \sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{64 \sqrt {2} a^{13/4}} \] Output:
-1/a^3/x-1/8*c*x^3/a^2/(c*x^4+a)^2-13/32*c*x^3/a^3/(c*x^4+a)-45/128*c^(1/4 )*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(13/4)-45/128*c^(1/4)*arc tan(1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(13/4)+45/128*c^(1/4)*arctanh(2 ^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))*2^(1/2)/a^(13/4)
Time = 0.08 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=\frac {-\frac {256 \sqrt [4]{a}}{x}-\frac {32 a^{5/4} c x^3}{\left (a+c x^4\right )^2}-\frac {104 \sqrt [4]{a} c x^3}{a+c x^4}+90 \sqrt {2} \sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-90 \sqrt {2} \sqrt [4]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-45 \sqrt {2} \sqrt [4]{c} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+45 \sqrt {2} \sqrt [4]{c} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{256 a^{13/4}} \] Input:
Integrate[1/(x^2*(a + c*x^4)^3),x]
Output:
((-256*a^(1/4))/x - (32*a^(5/4)*c*x^3)/(a + c*x^4)^2 - (104*a^(1/4)*c*x^3) /(a + c*x^4) + 90*Sqrt[2]*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 90*Sqrt[2]*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 45*Sqrt[2]* c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 45*Sqrt[2 ]*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(256*a^( 13/4))
Time = 0.76 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.48, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {819, 819, 847, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {9 \int \frac {1}{x^2 \left (c x^4+a\right )^2}dx}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{x^2 \left (c x^4+a\right )}dx}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \int \frac {x^2}{c x^4+a}dx}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {1}{a x}\right )}{4 a}+\frac {1}{4 a x \left (a+c x^4\right )}\right )}{8 a}+\frac {1}{8 a x \left (a+c x^4\right )^2}\) |
Input:
Int[1/(x^2*(a + c*x^4)^3),x]
Output:
1/(8*a*x*(a + c*x^4)^2) + (9*(1/(4*a*x*(a + c*x^4)) + (5*(-(1/(a*x)) - (c* ((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + A rcTan[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])))/a))/(4*a)))/(8*a)
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 0.50 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {-\frac {45 c^{2} x^{8}}{32 a^{3}}-\frac {81 c \,x^{4}}{32 a^{2}}-\frac {1}{a}}{x \left (c \,x^{4}+a \right )^{2}}+\frac {45 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{4}+c \right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4} a^{13}+4 c \right ) x +a^{10} \textit {\_R}^{3}\right )\right )}{128}\) | \(81\) |
| default | \(-\frac {1}{a^{3} x}-\frac {c \left (\frac {\frac {13}{32} c \,x^{7}+\frac {17}{32} a \,x^{3}}{\left (c \,x^{4}+a \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(141\) |
Input:
int(1/x^2/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(-45/32*c^2/a^3*x^8-81/32/a^2*c*x^4-1/a)/x/(c*x^4+a)^2+45/128*sum(_R*ln((5 *_R^4*a^13+4*c)*x+a^10*_R^3),_R=RootOf(_Z^4*a^13+c))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=-\frac {180 \, c^{2} x^{8} + 324 \, a c x^{4} + 45 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} \left (-\frac {c}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, c x\right ) + 45 \, {\left (-i \, a^{3} c^{2} x^{9} - 2 i \, a^{4} c x^{5} - i \, a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 i \, a^{10} \left (-\frac {c}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, c x\right ) + 45 \, {\left (i \, a^{3} c^{2} x^{9} + 2 i \, a^{4} c x^{5} + i \, a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 i \, a^{10} \left (-\frac {c}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, c x\right ) - 45 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} \left (-\frac {c}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, c x\right ) + 128 \, a^{2}}{128 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )}} \] Input:
integrate(1/x^2/(c*x^4+a)^3,x, algorithm="fricas")
Output:
-1/128*(180*c^2*x^8 + 324*a*c*x^4 + 45*(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x) *(-c/a^13)^(1/4)*log(91125*a^10*(-c/a^13)^(3/4) + 91125*c*x) + 45*(-I*a^3* c^2*x^9 - 2*I*a^4*c*x^5 - I*a^5*x)*(-c/a^13)^(1/4)*log(91125*I*a^10*(-c/a^ 13)^(3/4) + 91125*c*x) + 45*(I*a^3*c^2*x^9 + 2*I*a^4*c*x^5 + I*a^5*x)*(-c/ a^13)^(1/4)*log(-91125*I*a^10*(-c/a^13)^(3/4) + 91125*c*x) - 45*(a^3*c^2*x ^9 + 2*a^4*c*x^5 + a^5*x)*(-c/a^13)^(1/4)*log(-91125*a^10*(-c/a^13)^(3/4) + 91125*c*x) + 128*a^2)/(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=\frac {- 32 a^{2} - 81 a c x^{4} - 45 c^{2} x^{8}}{32 a^{5} x + 64 a^{4} c x^{5} + 32 a^{3} c^{2} x^{9}} + \operatorname {RootSum} {\left (268435456 t^{4} a^{13} + 4100625 c, \left ( t \mapsto t \log {\left (- \frac {2097152 t^{3} a^{10}}{91125 c} + x \right )} \right )\right )} \] Input:
integrate(1/x**2/(c*x**4+a)**3,x)
Output:
(-32*a**2 - 81*a*c*x**4 - 45*c**2*x**8)/(32*a**5*x + 64*a**4*c*x**5 + 32*a **3*c**2*x**9) + RootSum(268435456*_t**4*a**13 + 4100625*c, Lambda(_t, _t* log(-2097152*_t**3*a**10/(91125*c) + x)))
Time = 0.12 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=-\frac {45 \, c^{2} x^{8} + 81 \, a c x^{4} + 32 \, a^{2}}{32 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )}} - \frac {45 \, c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{256 \, a^{3}} \] Input:
integrate(1/x^2/(c*x^4+a)^3,x, algorithm="maxima")
Output:
-1/32*(45*c^2*x^8 + 81*a*c*x^4 + 32*a^2)/(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5* x) - 45/256*c*(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*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)) - sqrt(2)*log(sqrt(c)*x^2 + sqr t(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)))/a^3
Time = 0.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=-\frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{4} c^{2}} - \frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{4} c^{2}} + \frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{4} c^{2}} - \frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{4} c^{2}} - \frac {13 \, c^{2} x^{7} + 17 \, a c x^{3}}{32 \, {\left (c x^{4} + a\right )}^{2} a^{3}} - \frac {1}{a^{3} x} \] Input:
integrate(1/x^2/(c*x^4+a)^3,x, algorithm="giac")
Output:
-45/128*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4 ))/(a/c)^(1/4))/(a^4*c^2) - 45/128*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2 )*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^4*c^2) + 45/256*sqrt(2)*(a*c ^3)^(3/4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^4*c^2) - 45/256* sqrt(2)*(a*c^3)^(3/4)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^4*c^ 2) - 1/32*(13*c^2*x^7 + 17*a*c*x^3)/((c*x^4 + a)^2*a^3) - 1/(a^3*x)
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx=\frac {45\,{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{64\,a^{13/4}}-\frac {45\,{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{64\,a^{13/4}}-\frac {\frac {1}{a}+\frac {81\,c\,x^4}{32\,a^2}+\frac {45\,c^2\,x^8}{32\,a^3}}{a^2\,x+2\,a\,c\,x^5+c^2\,x^9} \] Input:
int(1/(x^2*(a + c*x^4)^3),x)
Output:
(45*(-c)^(1/4)*atanh(((-c)^(1/4)*x)/a^(1/4)))/(64*a^(13/4)) - (45*(-c)^(1/ 4)*atan(((-c)^(1/4)*x)/a^(1/4)))/(64*a^(13/4)) - (1/a + (81*c*x^4)/(32*a^2 ) + (45*c^2*x^8)/(32*a^3))/(a^2*x + c^2*x^9 + 2*a*c*x^5)
Time = 0.22 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.66 \[ \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(c*x^4+a)^3,x)
Output:
(90*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)))*a**2*x + 180*c**(1/4)*a**(3/4)*sqrt(2)*ata n((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a *c*x**5 + 90*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)))*c**2*x**9 - 90*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)))*a**2*x - 180*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*s qrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*x**5 - 90*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)))*c**2*x**9 - 45*c**(1/4)*a**(3/4)*sqrt(2)*log( - c**(1/4 )*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a**2*x - 90*c**(1/4)*a**(3/ 4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a* c*x**5 - 45*c**(1/4)*a**(3/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*c**2*x**9 + 45*c**(1/4)*a**(3/4)*sqrt(2)*log(c**( 1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a**2*x + 90*c**(1/4)*a** (3/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a* c*x**5 + 45*c**(1/4)*a**(3/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sq rt(a) + sqrt(c)*x**2)*c**2*x**9 - 256*a**3 - 648*a**2*c*x**4 - 360*a*c**2* x**8)/(256*a**4*x*(a**2 + 2*a*c*x**4 + c**2*x**8))