Integrand size = 17, antiderivative size = 187 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {d x}{c}+\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}} \] Output:
d*x/c-1/4*(a^(1/2)*d-c^(1/2)*e)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/ 2)/a^(1/4)/c^(5/4)-1/4*(a^(1/2)*d-c^(1/2)*e)*arctan(1+2^(1/2)*c^(1/4)*x/a^ (1/4))*2^(1/2)/a^(1/4)/c^(5/4)-1/4*(a^(1/2)*d+c^(1/2)*e)*arctanh(2^(1/2)*a ^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))*2^(1/2)/a^(1/4)/c^(5/4)
Time = 0.11 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.57 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {d x}{c}+\frac {\left (-a^{5/4} \sqrt {c} d+a^{3/4} c e\right ) \arctan \left (\frac {-\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{a}}\right )}{2 \sqrt {2} a c^{7/4}}+\frac {\left (-a^{5/4} \sqrt {c} d+a^{3/4} c e\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{a}}\right )}{2 \sqrt {2} a c^{7/4}}+\frac {\left (a^{5/4} \sqrt {c} d+a^{3/4} c e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a c^{7/4}}-\frac {\left (a^{5/4} \sqrt {c} d+a^{3/4} c e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a c^{7/4}} \] Input:
Integrate[(d + e/x^2)/(c + a/x^4),x]
Output:
(d*x)/c + ((-(a^(5/4)*Sqrt[c]*d) + a^(3/4)*c*e)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*c^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*a*c^(7/4)) + ((-(a^(5/4)*Sq rt[c]*d) + a^(3/4)*c*e)*ArcTan[(Sqrt[2]*a^(1/4) + 2*c^(1/4)*x)/(Sqrt[2]*a^ (1/4))])/(2*Sqrt[2]*a*c^(7/4)) + ((a^(5/4)*Sqrt[c]*d + a^(3/4)*c*e)*Log[Sq rt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a*c^(7/4)) - ((a^(5/4)*Sqrt[c]*d + a^(3/4)*c*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a*c^(7/4))
Time = 0.49 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.29, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {1728, 1603, 1482, 27, 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.
| \(\displaystyle \int \frac {d+\frac {e}{x^2}}{\frac {a}{x^4}+c} \, dx\) |
| \(\Big \downarrow \) 1728 |
| \(\displaystyle \int \frac {x^2 \left (d x^2+e\right )}{a+c x^4}dx\) |
| \(\Big \downarrow \) 1603 |
| \(\displaystyle \frac {d x}{c}-\frac {\int \frac {a d-c e x^2}{c x^4+a}dx}{c}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}{c x^4+a}dx+\frac {1}{2} \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}{c x^4+a}dx}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx+\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}\right )+\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx+\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx+\frac {1}{2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )+\frac {1}{2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )+\frac {1}{2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}\right )+\frac {1}{2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {1}{2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right )+\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{c}\) |
Input:
Int[(d + e/x^2)/(c + a/x^4),x]
Output:
(d*x)/c - ((Sqrt[c]*((Sqrt[a]*d)/Sqrt[c] - e)*(-(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]*((Sqrt[a]*d)/Sqrt[c] + e)*(-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)/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_) + (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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_ Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ [m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[ m])
Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> Int[x^(n*(2*p + q))*(e + d/x^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.24
| method | result | size |
| risch | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c^{2}}\) | \(45\) |
| default | \(\frac {d x}{c}+\frac {-\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c}\) | \(211\) |
Input:
int((d+e/x^2)/(c+a/x^4),x,method=_RETURNVERBOSE)
Output:
d*x/c+1/4/c^2*sum((_R^2*c*e-a*d)/_R^3*ln(x-_R),_R=RootOf(_Z^4*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (128) = 256\).
Time = 0.10 (sec) , antiderivative size = 754, normalized size of antiderivative = 4.03 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {c \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x + {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x - {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x + {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + c \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x - {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + 4 \, d x}{4 \, c} \] Input:
integrate((d+e/x^2)/(c+a/x^4),x, algorithm="fricas")
Output:
1/4*(c*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d* e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x + (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e ^2 + c^2*e^4)/(a*c^5)) + a^2*c*d^3 - a*c^2*d*e^2)*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)) - c*sqrt((c^2*sqrt(-(a ^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)*log(-(a^2*d^4 - c ^2*e^4)*x - (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + a^2*c*d^3 - a*c^2*d*e^2)*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^ 4)/(a*c^5)) + 2*d*e)/c^2)) - c*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x + (a*c^4*e*sqr t(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - a^2*c*d^3 + a*c^2*d*e^2) *sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^ 2)) + c*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2* d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x - (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2 *e^2 + c^2*e^4)/(a*c^5)) - a^2*c*d^3 + a*c^2*d*e^2)*sqrt(-(c^2*sqrt(-(a^2* d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)) + 4*d*x)/c
Time = 0.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.58 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\operatorname {RootSum} {\left (256 t^{4} a c^{5} - 64 t^{2} a c^{3} d e + a^{2} d^{4} + 2 a c d^{2} e^{2} + c^{2} e^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a c^{4} e - 4 t a^{2} c d^{3} + 12 t a c^{2} d e^{2}}{a^{2} d^{4} - c^{2} e^{4}} \right )} \right )\right )} + \frac {d x}{c} \] Input:
integrate((d+e/x**2)/(c+a/x**4),x)
Output:
RootSum(256*_t**4*a*c**5 - 64*_t**2*a*c**3*d*e + a**2*d**4 + 2*a*c*d**2*e* *2 + c**2*e**4, Lambda(_t, _t*log(x + (-64*_t**3*a*c**4*e - 4*_t*a**2*c*d* *3 + 12*_t*a*c**2*d*e**2)/(a**2*d**4 - c**2*e**4)))) + d*x/c
Time = 0.11 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.28 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {d x}{c} - \frac {\frac {2 \, \sqrt {2} {\left (a \sqrt {c} d - \sqrt {a} c e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (a \sqrt {c} d - \sqrt {a} c e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (a \sqrt {c} d + \sqrt {a} c e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (a \sqrt {c} d + \sqrt {a} c e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c} \] Input:
integrate((d+e/x^2)/(c+a/x^4),x, algorithm="maxima")
Output:
d*x/c - 1/8*(2*sqrt(2)*(a*sqrt(c)*d - sqrt(a)*c*e)*arctan(1/2*sqrt(2)*(2*s qrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(s qrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(a*sqrt(c)*d - sqrt(a)*c*e)*arctan(1/ 2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/( sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(a*sqrt(c)*d + sqrt(a)*c* e)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4) ) - sqrt(2)*(a*sqrt(c)*d + sqrt(a)*c*e)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)* c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/c
Time = 0.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.30 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {d x}{c} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \] Input:
integrate((d+e/x^2)/(c+a/x^4),x, algorithm="giac")
Output:
d*x/c - 1/4*sqrt(2)*((a*c^3)^(1/4)*a*c*d - (a*c^3)^(3/4)*e)*arctan(1/2*sqr t(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) - 1/4*sqrt(2)*((a*c^ 3)^(1/4)*a*c*d - (a*c^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^ (1/4))/(a/c)^(1/4))/(a*c^3) - 1/8*sqrt(2)*((a*c^3)^(1/4)*a*c*d + (a*c^3)^( 3/4)*e)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) + 1/8*sqrt(2) *((a*c^3)^(1/4)*a*c*d + (a*c^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3)
Time = 20.35 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.97 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {d\,x}{c}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,c\,d^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {d\,e}{8\,c^2}-\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}-\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {d\,e}{8\,c^2}-\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}-\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {-a\,c^5}-c\,e^2\,\sqrt {-a\,c^5}+2\,a\,c^3\,d\,e}{16\,a\,c^5}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,c\,d^2\,x\,\sqrt {\frac {d\,e}{8\,c^2}-\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}+\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d\,e}{8\,c^2}-\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}+\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}\right )\,\sqrt {\frac {c\,e^2\,\sqrt {-a\,c^5}-a\,d^2\,\sqrt {-a\,c^5}+2\,a\,c^3\,d\,e}{16\,a\,c^5}} \] Input:
int((d + e/x^2)/(c + a/x^4),x)
Output:
(d*x)/c - 2*atanh((8*a^2*c*d^2*x*((d^2*(-a*c^5)^(1/2))/(16*c^5) + (d*e)/(8 *c^2) - (e^2*(-a*c^5)^(1/2))/(16*a*c^4))^(1/2))/(2*a^2*d^2*e - 2*a*c*e^3 + (2*a^2*d^3*(-a*c^5)^(1/2))/c^3 - (2*a*d*e^2*(-a*c^5)^(1/2))/c^2) - (8*a*c ^2*e^2*x*((d^2*(-a*c^5)^(1/2))/(16*c^5) + (d*e)/(8*c^2) - (e^2*(-a*c^5)^(1 /2))/(16*a*c^4))^(1/2))/(2*a^2*d^2*e - 2*a*c*e^3 + (2*a^2*d^3*(-a*c^5)^(1/ 2))/c^3 - (2*a*d*e^2*(-a*c^5)^(1/2))/c^2))*((a*d^2*(-a*c^5)^(1/2) - c*e^2* (-a*c^5)^(1/2) + 2*a*c^3*d*e)/(16*a*c^5))^(1/2) - 2*atanh((8*a^2*c*d^2*x*( (d*e)/(8*c^2) - (d^2*(-a*c^5)^(1/2))/(16*c^5) + (e^2*(-a*c^5)^(1/2))/(16*a *c^4))^(1/2))/(2*a^2*d^2*e - 2*a*c*e^3 - (2*a^2*d^3*(-a*c^5)^(1/2))/c^3 + (2*a*d*e^2*(-a*c^5)^(1/2))/c^2) - (8*a*c^2*e^2*x*((d*e)/(8*c^2) - (d^2*(-a *c^5)^(1/2))/(16*c^5) + (e^2*(-a*c^5)^(1/2))/(16*a*c^4))^(1/2))/(2*a^2*d^2 *e - 2*a*c*e^3 - (2*a^2*d^3*(-a*c^5)^(1/2))/c^3 + (2*a*d*e^2*(-a*c^5)^(1/2 ))/c^2))*((c*e^2*(-a*c^5)^(1/2) - a*d^2*(-a*c^5)^(1/2) + 2*a*c^3*d*e)/(16* a*c^5))^(1/2)
Time = 0.20 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx=\frac {-2 c^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) e +2 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d +2 c^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) e -2 c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d +c^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) e -c^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) e +c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d -c^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d +8 a c d x}{8 a \,c^{2}} \] Input:
int((d+e/x^2)/(c+a/x^4),x)
Output:
( - 2*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*e + 2*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*d + 2*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*e - 2*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*d + c**(1/4)*a**(3/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + s qrt(c)*x**2)*c*e - 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*e + 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*d - c**(3/4)*a**(1/4)*sq rt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*d + 8*a* c*d*x)/(8*a*c**2)