Integrand size = 22, antiderivative size = 54 \[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 \sqrt {a} \sqrt {b c-a d}} \] Output:
1/2*arctan((-a*d+b*c)^(1/2)*x^2/a^(1/2)/(d*x^4+c)^(1/2))/a^(1/2)/(-a*d+b*c )^(1/2)
Time = 0.54 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.37 \[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 \sqrt {a} \sqrt {b c-a d}} \] Input:
Integrate[x/((a + b*x^4)*Sqrt[c + d*x^4]),x]
Output:
ArcTan[(a*Sqrt[d] + b*x^2*(Sqrt[d]*x^2 + Sqrt[c + d*x^4]))/(Sqrt[a]*Sqrt[b *c - a*d])]/(2*Sqrt[a]*Sqrt[b*c - a*d])
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {965, 291, 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.
\(\displaystyle \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{2} \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 \sqrt {a} \sqrt {b c-a d}}\) |
Input:
Int[x/((a + b*x^4)*Sqrt[c + d*x^4]),x]
Output:
ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])]/(2*Sqrt[a]*Sqrt[b* c - a*d])
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/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; 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 = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{2 \sqrt {a \left (a d -c b \right )}}\) | \(42\) |
default | \(-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\) | \(322\) |
elliptic | \(-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\) | \(322\) |
Input:
int(x/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/(a*(a*d-b*c))^(1/2)*arctanh(a*(d*x^4+c)^(1/2)/x^2/(a*(a*d-b*c))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (42) = 84\).
Time = 0.14 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.54 \[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [-\frac {\sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, {\left (a b c - a^{2} d\right )}}, \frac {\arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right )}{4 \, \sqrt {a b c - a^{2} d}}\right ] \] Input:
integrate(x/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
[-1/8*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2* (3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 - 4*((b*c - 2*a*d)*x^6 - a*c*x^2)*sq rt(d*x^4 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + a^2))/(a*b*c - a^2*d), 1/4*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt(a*b* c - a^2*d)/((a*b*c*d - a^2*d^2)*x^6 + (a*b*c^2 - a^2*c*d)*x^2))/sqrt(a*b*c - a^2*d)]
\[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(x/(b*x**4+a)/(d*x**4+c)**(1/2),x)
Output:
Integral(x/((a + b*x**4)*sqrt(c + d*x**4)), x)
\[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {x}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c}} \,d x } \] Input:
integrate(x/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
Output:
integrate(x/((b*x^4 + a)*sqrt(d*x^4 + c)), x)
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}}} \] Input:
integrate(x/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
-1/2*sqrt(d)*arctan(1/2*((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b - b*c + 2*a*d )/sqrt(a*b*c*d - a^2*d^2))/sqrt(a*b*c*d - a^2*d^2)
Timed out. \[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x}{\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \] Input:
int(x/((a + b*x^4)*(c + d*x^4)^(1/2)),x)
Output:
int(x/((a + b*x^4)*(c + d*x^4)^(1/2)), x)
Time = 0.24 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.06 \[ \int \frac {x}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\sqrt {a}\, \sqrt {a d -b c}\, \left (\mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right )+\mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right )-\mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b \,x^{2}+2 a d +2 b d \,x^{4}\right )\right )}{4 a \left (a d -b c \right )} \] Input:
int(x/(b*x^4+a)/(d*x^4+c)^(1/2),x)
Output:
(sqrt(a)*sqrt(a*d - b*c)*(log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2) + log(sqrt (2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x** 4) + sqrt(d)*sqrt(b)*x**2) - log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqr t(d)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x**4)))/(4*a*(a*d - b*c))