Integrand size = 17, antiderivative size = 245 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=-\frac {(b c-a d) x}{4 c d \left (c+d x^4\right )}-\frac {(b c+3 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c+3 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{5/4}}+\frac {(b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{5/4}} \]
-1/4*(-a*d+b*c)*x/c/d/(d*x^4+c)+1/16*(3*a*d+b*c)*arctan(-1+d^(1/4)*x*2^(1/ 2)/c^(1/4))/c^(7/4)/d^(5/4)*2^(1/2)+1/16*(3*a*d+b*c)*arctan(1+d^(1/4)*x*2^ (1/2)/c^(1/4))/c^(7/4)/d^(5/4)*2^(1/2)-1/32*(3*a*d+b*c)*ln(-c^(1/4)*d^(1/4 )*x*2^(1/2)+c^(1/2)+x^2*d^(1/2))/c^(7/4)/d^(5/4)*2^(1/2)+1/32*(3*a*d+b*c)* ln(c^(1/4)*d^(1/4)*x*2^(1/2)+c^(1/2)+x^2*d^(1/2))/c^(7/4)/d^(5/4)*2^(1/2)
Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=\frac {-\frac {8 c^{3/4} \sqrt [4]{d} (b c-a d) x}{c+d x^4}-2 \sqrt {2} (b c+3 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt {2} (b c+3 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-\sqrt {2} (b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )+\sqrt {2} (b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{32 c^{7/4} d^{5/4}} \]
((-8*c^(3/4)*d^(1/4)*(b*c - a*d)*x)/(c + d*x^4) - 2*Sqrt[2]*(b*c + 3*a*d)* ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[2]*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - Sqrt[2]*(b*c + 3*a*d)*Log[Sqrt[c] - Sqrt [2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*(b*c + 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(32*c^(7/4)*d^(5/4))
Time = 0.42 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {910, 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.
| \(\displaystyle \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {(3 a d+b c) \int \frac {1}{d x^4+c}dx}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x^2+\sqrt {c}}{d x^4+c}dx}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {d}}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{d x^4+c}dx}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{\sqrt [4]{d} \left (x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} x}{x^2-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} x+\sqrt [4]{c}}{x^2+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}dx}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(3 a d+b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{4 c d}-\frac {x (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
-1/4*((b*c - a*d)*x)/(c*d*(c + d*x^4)) + ((b*c + 3*a*d)*((-(ArcTan[1 - (Sq rt[2]*d^(1/4)*x)/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2] *d^(1/4)*x)/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c]) + (-1/2*Log[Sq rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2]/(2*Sqrt[2]*c^(1/ 4)*d^(1/4)))/(2*Sqrt[c])))/(4*c*d)
3.2.51.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.27
| method | result | size |
| risch | \(\frac {\left (a d -b c \right ) x}{4 d c \left (d \,x^{4}+c \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {\left (3 a d +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 c \,d^{2}}\) | \(65\) |
| default | \(\frac {\left (a d -b c \right ) x}{4 d c \left (d \,x^{4}+c \right )}+\frac {\left (3 a d +b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2} d}\) | \(140\) |
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.64 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=\frac {{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (c^{2} d \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} + {\left (b c + 3 \, a d\right )} x\right ) - {\left (-i \, c d^{2} x^{4} - i \, c^{2} d\right )} \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (i \, c^{2} d \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} + {\left (b c + 3 \, a d\right )} x\right ) - {\left (i \, c d^{2} x^{4} + i \, c^{2} d\right )} \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (-i \, c^{2} d \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} + {\left (b c + 3 \, a d\right )} x\right ) - {\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (-c^{2} d \left (-\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac {1}{4}} + {\left (b c + 3 \, a d\right )} x\right ) - 4 \, {\left (b c - a d\right )} x}{16 \, {\left (c d^{2} x^{4} + c^{2} d\right )}} \]
1/16*((c*d^2*x^4 + c^2*d)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(c^7*d^5))^(1/4)*log(c^2*d*(-(b^4*c^4 + 1 2*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(c^7*d^ 5))^(1/4) + (b*c + 3*a*d)*x) - (-I*c*d^2*x^4 - I*c^2*d)*(-(b^4*c^4 + 12*a* b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(c^7*d^5))^ (1/4)*log(I*c^2*d*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a ^3*b*c*d^3 + 81*a^4*d^4)/(c^7*d^5))^(1/4) + (b*c + 3*a*d)*x) - (I*c*d^2*x^ 4 + I*c^2*d)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b* c*d^3 + 81*a^4*d^4)/(c^7*d^5))^(1/4)*log(-I*c^2*d*(-(b^4*c^4 + 12*a*b^3*c^ 3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(c^7*d^5))^(1/4) + (b*c + 3*a*d)*x) - (c*d^2*x^4 + c^2*d)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54* a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(c^7*d^5))^(1/4)*log(-c^2* d*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81* a^4*d^4)/(c^7*d^5))^(1/4) + (b*c + 3*a*d)*x) - 4*(b*c - a*d)*x)/(c*d^2*x^4 + c^2*d)
Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.46 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=\frac {x \left (a d - b c\right )}{4 c^{2} d + 4 c d^{2} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} c^{7} d^{5} + 81 a^{4} d^{4} + 108 a^{3} b c d^{3} + 54 a^{2} b^{2} c^{2} d^{2} + 12 a b^{3} c^{3} d + b^{4} c^{4}, \left ( t \mapsto t \log {\left (\frac {16 t c^{2} d}{3 a d + b c} + x \right )} \right )\right )} \]
x*(a*d - b*c)/(4*c**2*d + 4*c*d**2*x**4) + RootSum(65536*_t**4*c**7*d**5 + 81*a**4*d**4 + 108*a**3*b*c*d**3 + 54*a**2*b**2*c**2*d**2 + 12*a*b**3*c** 3*d + b**4*c**4, Lambda(_t, _t*log(16*_t*c**2*d/(3*a*d + b*c) + x)))
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=-\frac {{\left (b c - a d\right )} x}{4 \, {\left (c d^{2} x^{4} + c^{2} d\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b c + 3 \, a d\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c + 3 \, a d\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{32 \, c d} \]
-1/4*(b*c - a*d)*x/(c*d^2*x^4 + c^2*d) + 1/32*(2*sqrt(2)*(b*c + 3*a*d)*arc tan(1/2*sqrt(2)*(2*sqrt(d)*x + sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt( d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(b*c + 3*a*d)*arctan(1/2* sqrt(2)*(2*sqrt(d)*x - sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sq rt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(b*c + 3*a*d)*log(sqrt(d)*x^2 + sqr t(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(b*c + 3*a*d )*log(sqrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)) )/(c*d)
Time = 0.28 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.09 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{16 \, c^{2} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{16 \, c^{2} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{32 \, c^{2} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{32 \, c^{2} d^{2}} - \frac {b c x - a d x}{4 \, {\left (d x^{4} + c\right )} c d} \]
1/16*sqrt(2)*((c*d^3)^(1/4)*b*c + 3*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)* (2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c^2*d^2) + 1/16*sqrt(2)*((c*d^3) ^(1/4)*b*c + 3*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d)^ (1/4))/(c/d)^(1/4))/(c^2*d^2) + 1/32*sqrt(2)*((c*d^3)^(1/4)*b*c + 3*(c*d^3 )^(1/4)*a*d)*log(x^2 + sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c^2*d^2) - 1/32 *sqrt(2)*((c*d^3)^(1/4)*b*c + 3*(c*d^3)^(1/4)*a*d)*log(x^2 - sqrt(2)*x*(c/ d)^(1/4) + sqrt(c/d))/(c^2*d^2) - 1/4*(b*c*x - a*d*x)/((d*x^4 + c)*c*d)
Time = 5.69 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.02 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}-\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}+\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}+\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}}{\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}-\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}-\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}+\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}}\right )\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-c\right )}^{7/4}\,d^{5/4}}+\frac {x\,\left (a\,d-b\,c\right )}{4\,c\,d\,\left (d\,x^4+c\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}-\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}+\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}+\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}}{\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}-\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}-\frac {\left (\frac {x\,\left (9\,a^2\,d^3+6\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{4\,c^2}+\frac {\left (3\,a\,d+b\,c\right )\,\left (12\,a\,d^3+4\,b\,c\,d^2\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}\right )\,\left (3\,a\,d+b\,c\right )}{16\,{\left (-c\right )}^{7/4}\,d^{5/4}}}\right )\,\left (3\,a\,d+b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{5/4}} \]
(atan(((((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d^2))/(4*c^2) - ((3*a*d + b*c )*(12*a*d^3 + 4*b*c*d^2))/(16*(-c)^(7/4)*d^(5/4)))*(3*a*d + b*c)*1i)/(16*( -c)^(7/4)*d^(5/4)) + (((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d^2))/(4*c^2) + ((3*a*d + b*c)*(12*a*d^3 + 4*b*c*d^2))/(16*(-c)^(7/4)*d^(5/4)))*(3*a*d + b*c)*1i)/(16*(-c)^(7/4)*d^(5/4)))/((((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d ^2))/(4*c^2) - ((3*a*d + b*c)*(12*a*d^3 + 4*b*c*d^2))/(16*(-c)^(7/4)*d^(5/ 4)))*(3*a*d + b*c))/(16*(-c)^(7/4)*d^(5/4)) - (((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d^2))/(4*c^2) + ((3*a*d + b*c)*(12*a*d^3 + 4*b*c*d^2))/(16*(-c)^ (7/4)*d^(5/4)))*(3*a*d + b*c))/(16*(-c)^(7/4)*d^(5/4))))*(3*a*d + b*c)*1i) /(8*(-c)^(7/4)*d^(5/4)) + (atan(((((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d^2 ))/(4*c^2) - ((3*a*d + b*c)*(12*a*d^3 + 4*b*c*d^2)*1i)/(16*(-c)^(7/4)*d^(5 /4)))*(3*a*d + b*c))/(16*(-c)^(7/4)*d^(5/4)) + (((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d^2))/(4*c^2) + ((3*a*d + b*c)*(12*a*d^3 + 4*b*c*d^2)*1i)/(16*( -c)^(7/4)*d^(5/4)))*(3*a*d + b*c))/(16*(-c)^(7/4)*d^(5/4)))/((((x*(9*a^2*d ^3 + b^2*c^2*d + 6*a*b*c*d^2))/(4*c^2) - ((3*a*d + b*c)*(12*a*d^3 + 4*b*c* d^2)*1i)/(16*(-c)^(7/4)*d^(5/4)))*(3*a*d + b*c)*1i)/(16*(-c)^(7/4)*d^(5/4) ) - (((x*(9*a^2*d^3 + b^2*c^2*d + 6*a*b*c*d^2))/(4*c^2) + ((3*a*d + b*c)*( 12*a*d^3 + 4*b*c*d^2)*1i)/(16*(-c)^(7/4)*d^(5/4)))*(3*a*d + b*c)*1i)/(16*( -c)^(7/4)*d^(5/4))))*(3*a*d + b*c))/(8*(-c)^(7/4)*d^(5/4)) + (x*(a*d - b*c ))/(4*c*d*(c + d*x^4))