Integrand size = 24, antiderivative size = 85 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{2 a}+\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a \sqrt {b}} \]
-1/2*arctanh((d*x^4+c)^(1/2)/c^(1/2))*c^(1/2)/a+1/2*arctanh(b^(1/2)*(d*x^4 +c)^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^(1/2)/a/b^(1/2)
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\frac {\frac {\sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{\sqrt {b}}-\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{2 a} \]
((Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]]) /Sqrt[b] - Sqrt[c]*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/(2*a)
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {948, 94, 73, 221}
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.
| \(\displaystyle \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{4} \int \frac {\sqrt {d x^4+c}}{x^4 \left (b x^4+a\right )}dx^4\) |
| \(\Big \downarrow \) 94 |
| \(\displaystyle \frac {1}{4} \left (\frac {c \int \frac {1}{x^4 \sqrt {d x^4+c}}dx^4}{a}-\frac {(b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{a}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 c \int \frac {1}{\frac {x^8}{d}-\frac {c}{d}}d\sqrt {d x^4+c}}{a d}-\frac {2 (b c-a d) \int \frac {1}{\frac {b x^8}{d}+a-\frac {b c}{d}}d\sqrt {d x^4+c}}{a d}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{a}\right )\) |
((-2*Sqrt[c]*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/a + (2*Sqrt[b*c - a*d]*ArcT anh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/4
3.8.91.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[(b*e - a*f)/(b*c - a*d) Int[(e + f*x)^(p - 1)/(a + b*x), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[(e + f*x)^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 4.87 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24
| method | result | size |
| pseudoelliptic | \(\frac {\arctan \left (\frac {b \sqrt {d \,x^{4}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a d -\arctan \left (\frac {b \sqrt {d \,x^{4}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) b c -\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{\sqrt {c}}\right ) \sqrt {\left (a d -b c \right ) b}}{2 a \sqrt {\left (a d -b c \right ) b}}\) | \(105\) |
| elliptic | \(\frac {\sqrt {d \,x^{4}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{2 a}-\frac {\sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}}{4 a}-\frac {\sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}}{4 a}\) | \(726\) |
| default | \(\frac {\frac {\sqrt {d \,x^{4}+c}}{2}-\frac {\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{2}}{a}-\frac {b \left (\frac {\sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}}{4 b}+\frac {\sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}}{4 b}\right )}{a}\) | \(734\) |
1/2*(arctan(b*(d*x^4+c)^(1/2)/((a*d-b*c)*b)^(1/2))*a*d-arctan(b*(d*x^4+c)^ (1/2)/((a*d-b*c)*b)^(1/2))*b*c-c^(1/2)*arctanh((d*x^4+c)^(1/2)/c^(1/2))*(( a*d-b*c)*b)^(1/2))/a/((a*d-b*c)*b)^(1/2)
Time = 0.27 (sec) , antiderivative size = 383, normalized size of antiderivative = 4.51 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\left [\frac {\sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right ) + \sqrt {c} \log \left (\frac {d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right )}{4 \, a}, \frac {2 \, \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + \sqrt {c} \log \left (\frac {d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right )}{4 \, a}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-c}}{c}\right ) + \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right )}{4 \, a}, \frac {\sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-c}}{c}\right )}{2 \, a}\right ] \]
[1/4*(sqrt((b*c - a*d)/b)*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)*b *sqrt((b*c - a*d)/b))/(b*x^4 + a)) + sqrt(c)*log((d*x^4 - 2*sqrt(d*x^4 + c )*sqrt(c) + 2*c)/x^4))/a, 1/4*(2*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x^4 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + sqrt(c)*log((d*x^4 - 2*sqrt(d*x^ 4 + c)*sqrt(c) + 2*c)/x^4))/a, 1/4*(2*sqrt(-c)*arctan(sqrt(d*x^4 + c)*sqrt (-c)/c) + sqrt((b*c - a*d)/b)*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)*b*sqrt((b*c - a*d)/b))/(b*x^4 + a)))/a, 1/2*(sqrt(-(b*c - a*d)/b)*arcta n(-sqrt(d*x^4 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + sqrt(-c)*arctan(s qrt(d*x^4 + c)*sqrt(-c)/c))/a]
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (70) = 140\).
Time = 4.43 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\begin {cases} \frac {2 \left (\frac {c d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {- c}} \right )}}{4 a \sqrt {- c}} + \frac {d \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 a b \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (- \frac {b \left (\begin {cases} \frac {\frac {a}{2 b} + x^{4}}{a} & \text {for}\: b = 0 \\- \frac {\log {\left (a - 2 b \left (\frac {a}{2 b} + x^{4}\right ) \right )}}{2 b} & \text {otherwise} \end {cases}\right )}{2 a} - \frac {b \left (\begin {cases} \frac {\frac {a}{2 b} + x^{4}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + 2 b \left (\frac {a}{2 b} + x^{4}\right ) \right )}}{2 b} & \text {otherwise} \end {cases}\right )}{2 a}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(c*d*atan(sqrt(c + d*x**4)/sqrt(-c))/(4*a*sqrt(-c)) + d*(a*d - b*c)*atan(sqrt(c + d*x**4)/sqrt((a*d - b*c)/b))/(4*a*b*sqrt((a*d - b*c)/ b)))/d, Ne(d, 0)), (sqrt(c)*(-b*Piecewise(((a/(2*b) + x**4)/a, Eq(b, 0)), (-log(a - 2*b*(a/(2*b) + x**4))/(2*b), True))/(2*a) - b*Piecewise(((a/(2*b ) + x**4)/a, Eq(b, 0)), (log(a + 2*b*(a/(2*b) + x**4))/(2*b), True))/(2*a) ), True))
\[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=-\frac {{\left (b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a} + \frac {c \arctan \left (\frac {\sqrt {d x^{4} + c}}{\sqrt {-c}}\right )}{2 \, a \sqrt {-c}} \]
-1/2*(b*c - a*d)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2 *c + a*b*d)*a) + 1/2*c*arctan(sqrt(d*x^4 + c)/sqrt(-c))/(a*sqrt(-c))
Time = 9.24 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {c+d x^4}}{x \left (a+b x^4\right )} \, dx=\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c}\,\left (\sqrt {d\,x^4+c}\,\left (\frac {a^2\,b\,d^4}{2}-a\,b^2\,c\,d^3+b^3\,c^2\,d^2\right )+\frac {c\,\left (8\,a^3\,b^2\,d^3-16\,a^2\,b^3\,c\,d^2\right )\,\sqrt {d\,x^4+c}}{16\,a^2}\right )}{2\,a\,\left (\frac {b^2\,c^2\,d^3}{4}-\frac {a\,b\,c\,d^4}{4}\right )}\right )}{2\,a}+\frac {\mathrm {atanh}\left (\frac {a\,b^2\,c\,d^3\,\sqrt {d\,x^4+c}\,\sqrt {b^2\,c-a\,b\,d}}{4\,\left (\frac {a\,b^3\,c^2\,d^3}{4}-\frac {a^2\,b^2\,c\,d^4}{4}\right )}\right )\,\sqrt {b^2\,c-a\,b\,d}}{2\,a\,b} \]
(c^(1/2)*atanh((c^(1/2)*((c + d*x^4)^(1/2)*((a^2*b*d^4)/2 + b^3*c^2*d^2 - a*b^2*c*d^3) + (c*(8*a^3*b^2*d^3 - 16*a^2*b^3*c*d^2)*(c + d*x^4)^(1/2))/(1 6*a^2)))/(2*a*((b^2*c^2*d^3)/4 - (a*b*c*d^4)/4))))/(2*a) + (atanh((a*b^2*c *d^3*(c + d*x^4)^(1/2)*(b^2*c - a*b*d)^(1/2))/(4*((a*b^3*c^2*d^3)/4 - (a^2 *b^2*c*d^4)/4)))*(b^2*c - a*b*d)^(1/2))/(2*a*b)