Integrand size = 20, antiderivative size = 79 \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} (b c-a d)}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {c} (b c-a d)} \]
1/2*arctan(x^2*b^(1/2)/a^(1/2))*b^(1/2)/(-a*d+b*c)/a^(1/2)-1/2*arctan(x^2* d^(1/2)/c^(1/2))*d^(1/2)/(-a*d+b*c)/c^(1/2)
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {c}}}{2 b c-2 a d} \]
((Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[a] - (Sqrt[d]*ArcTan[(Sqrt[d ]*x^2)/Sqrt[c]])/Sqrt[c])/(2*b*c - 2*a*d)
Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {965, 303, 218}
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.
\(\displaystyle \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2\) |
\(\Big \downarrow \) 303 |
\(\displaystyle \frac {1}{2} \left (\frac {b \int \frac {1}{b x^4+a}dx^2}{b c-a d}-\frac {d \int \frac {1}{d x^4+c}dx^2}{b c-a d}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}\right )\) |
((Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)) - (Sqrt[d]* ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)))/2
3.8.76.3.1 Defintions of rubi rules used
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[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[b/(b *c - a*d) Int[1/(a + b*x^2), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x ^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 4.53 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {b \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -b c \right ) \sqrt {a b}}+\frac {d \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -b c \right ) \sqrt {c d}}\) | \(60\) |
risch | \(\frac {\sqrt {-c d}\, \ln \left (\left (-a c \,d^{3}+b \,c^{2} d^{2}\right ) x^{2}+\left (-c d \right )^{\frac {3}{2}} a d +\left (-c d \right )^{\frac {3}{2}} b c +2 \sqrt {-c d}\, b \,c^{2} d \right )}{4 c \left (a d -b c \right )}-\frac {\sqrt {-c d}\, \ln \left (\left (-a c \,d^{3}+b \,c^{2} d^{2}\right ) x^{2}-\left (-c d \right )^{\frac {3}{2}} a d -\left (-c d \right )^{\frac {3}{2}} b c -2 \sqrt {-c d}\, b \,c^{2} d \right )}{4 c \left (a d -b c \right )}+\frac {\sqrt {-a b}\, \ln \left (\left (-a^{2} b^{2} d +a \,b^{3} c \right ) x^{2}+\left (-a b \right )^{\frac {3}{2}} a d +\left (-a b \right )^{\frac {3}{2}} b c +2 \sqrt {-a b}\, a^{2} b d \right )}{4 a \left (a d -b c \right )}-\frac {\sqrt {-a b}\, \ln \left (\left (-a^{2} b^{2} d +a \,b^{3} c \right ) x^{2}-\left (-a b \right )^{\frac {3}{2}} a d -\left (-a b \right )^{\frac {3}{2}} b c -2 \sqrt {-a b}\, a^{2} b d \right )}{4 a \left (a d -b c \right )}\) | \(302\) |
-1/2*b/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))+1/2*d/(a*d-b*c)/(c* d)^(1/2)*arctan(d*x^2/(c*d)^(1/2))
Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.11 \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [-\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right )}{4 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {d}{c}} \arctan \left (\frac {c \sqrt {\frac {d}{c}}}{d x^{2}}\right ) - \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right )}{4 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b x^{2}}\right ) + \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right )}{4 \, {\left (b c - a d\right )}}, -\frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b x^{2}}\right ) - \sqrt {\frac {d}{c}} \arctan \left (\frac {c \sqrt {\frac {d}{c}}}{d x^{2}}\right )}{2 \, {\left (b c - a d\right )}}\right ] \]
[-1/4*(sqrt(-b/a)*log((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + sqrt (-d/c)*log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)))/(b*c - a*d), 1/4 *(2*sqrt(d/c)*arctan(c*sqrt(d/c)/(d*x^2)) - sqrt(-b/a)*log((b*x^4 - 2*a*x^ 2*sqrt(-b/a) - a)/(b*x^4 + a)))/(b*c - a*d), -1/4*(2*sqrt(b/a)*arctan(a*sq rt(b/a)/(b*x^2)) + sqrt(-d/c)*log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)))/(b*c - a*d), -1/2*(sqrt(b/a)*arctan(a*sqrt(b/a)/(b*x^2)) - sqrt(d/c )*arctan(c*sqrt(d/c)/(d*x^2)))/(b*c - a*d)]
Timed out. \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \]
1/2*b*arctan(b*x^2/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) - 1/2*d*arctan(d*x^2 /sqrt(c*d))/((b*c - a*d)*sqrt(c*d))
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \]
1/2*b*arctan(b*x^2/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) - 1/2*d*arctan(d*x^2 /sqrt(c*d))/((b*c - a*d)*sqrt(c*d))
Time = 10.41 (sec) , antiderivative size = 399, normalized size of antiderivative = 5.05 \[ \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\ln \left (a^2\,d^2\,{\left (-a\,b\right )}^{5/2}+b^2\,c^2\,{\left (-a\,b\right )}^{5/2}+2\,c\,d\,{\left (-a\,b\right )}^{7/2}-a^2\,b^5\,c^2\,x^2-a^4\,b^3\,d^2\,x^2+2\,a^3\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,a^2\,d-4\,a\,b\,c}-\frac {\ln \left (a^2\,d^2\,{\left (-a\,b\right )}^{5/2}+b^2\,c^2\,{\left (-a\,b\right )}^{5/2}+2\,c\,d\,{\left (-a\,b\right )}^{7/2}+a^2\,b^5\,c^2\,x^2+a^4\,b^3\,d^2\,x^2-2\,a^3\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,\left (a^2\,d-a\,b\,c\right )}-\frac {\ln \left (a^2\,d^2\,{\left (-c\,d\right )}^{5/2}+b^2\,c^2\,{\left (-c\,d\right )}^{5/2}+2\,a\,b\,{\left (-c\,d\right )}^{7/2}+a^2\,c^2\,d^5\,x^2+b^2\,c^4\,d^3\,x^2-2\,a\,b\,c^3\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,\left (b\,c^2-a\,c\,d\right )}+\frac {\ln \left (a^2\,d^2\,{\left (-c\,d\right )}^{5/2}+b^2\,c^2\,{\left (-c\,d\right )}^{5/2}+2\,a\,b\,{\left (-c\,d\right )}^{7/2}-a^2\,c^2\,d^5\,x^2-b^2\,c^4\,d^3\,x^2+2\,a\,b\,c^3\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,b\,c^2-4\,a\,c\,d} \]
(log(a^2*d^2*(-a*b)^(5/2) + b^2*c^2*(-a*b)^(5/2) + 2*c*d*(-a*b)^(7/2) - a^ 2*b^5*c^2*x^2 - a^4*b^3*d^2*x^2 + 2*a^3*b^4*c*d*x^2)*(-a*b)^(1/2))/(4*a^2* d - 4*a*b*c) - (log(a^2*d^2*(-a*b)^(5/2) + b^2*c^2*(-a*b)^(5/2) + 2*c*d*(- a*b)^(7/2) + a^2*b^5*c^2*x^2 + a^4*b^3*d^2*x^2 - 2*a^3*b^4*c*d*x^2)*(-a*b) ^(1/2))/(4*(a^2*d - a*b*c)) - (log(a^2*d^2*(-c*d)^(5/2) + b^2*c^2*(-c*d)^( 5/2) + 2*a*b*(-c*d)^(7/2) + a^2*c^2*d^5*x^2 + b^2*c^4*d^3*x^2 - 2*a*b*c^3* d^4*x^2)*(-c*d)^(1/2))/(4*(b*c^2 - a*c*d)) + (log(a^2*d^2*(-c*d)^(5/2) + b ^2*c^2*(-c*d)^(5/2) + 2*a*b*(-c*d)^(7/2) - a^2*c^2*d^5*x^2 - b^2*c^4*d^3*x ^2 + 2*a*b*c^3*d^4*x^2)*(-c*d)^(1/2))/(4*b*c^2 - 4*a*c*d)