Integrand size = 25, antiderivative size = 124 \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {\arctan \left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]
1/2*arctan(a^(3/4)*(1-(-b*x^2+a)^(1/2)/a^(1/2))/x/(-b*x^2+a)^(1/4)/b^(1/2) )/a^(3/4)/b^(1/2)+1/2*arctanh(a^(3/4)*(1+(-b*x^2+a)^(1/2)/a^(1/2))/x/(-b*x ^2+a)^(1/4)/b^(1/2))/a^(3/4)/b^(1/2)
Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {\arctan \left (\frac {b x^2-2 \sqrt {a} \sqrt {a-b x^2}}{2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a-b x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a-b x^2}}{b x^2+2 \sqrt {a} \sqrt {a-b x^2}}\right )}{4 a^{3/4} \sqrt {b}} \]
(ArcTan[(b*x^2 - 2*Sqrt[a]*Sqrt[a - b*x^2])/(2*a^(1/4)*Sqrt[b]*x*(a - b*x^ 2)^(1/4))] + ArcTanh[(2*a^(1/4)*Sqrt[b]*x*(a - b*x^2)^(1/4))/(b*x^2 + 2*Sq rt[a]*Sqrt[a - b*x^2])])/(4*a^(3/4)*Sqrt[b])
Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {308}
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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 308 |
| \(\displaystyle \frac {\arctan \left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}\) |
ArcTan[(a^(3/4)*(1 - Sqrt[a - b*x^2]/Sqrt[a]))/(Sqrt[b]*x*(a - b*x^2)^(1/4 ))]/(2*a^(3/4)*Sqrt[b]) + ArcTanh[(a^(3/4)*(1 + Sqrt[a - b*x^2]/Sqrt[a]))/ (Sqrt[b]*x*(a - b*x^2)^(1/4))]/(2*a^(3/4)*Sqrt[b])
3.4.11.3.1 Defintions of rubi rules used
Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[b^2/a, 4]}, Simp[(-b/(2*a*d*q))*ArcTan[(b + q^2*Sqrt[a + b*x^2])/ (q^3*x*(a + b*x^2)^(1/4))], x] - Simp[(b/(2*a*d*q))*ArcTanh[(b - q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ [b*c - 2*a*d, 0] && PosQ[b^2/a]
\[\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (-b \,x^{2}+2 a \right )}d x\]
Result contains complex when optimal does not.
Time = 25.30 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.70 \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {-2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) \]
1/4*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log(-(2*(1/4)^(3/4)*sqrt(-b*x^2 + a)* a^2*b^2*x*(-1/(a^3*b^2))^(3/4) + (-b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2 )) + (1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) - (-b*x^2 + a)^(3/4))/(b*x^2 - 2*a)) - 1/4*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log((2*(1/4)^(3/4)*sqrt(-b*x ^2 + a)*a^2*b^2*x*(-1/(a^3*b^2))^(3/4) - (-b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/ (a^3*b^2)) + (1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) + (-b*x^2 + a)^(3/4))/ (b*x^2 - 2*a)) + 1/4*I*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log((2*I*(1/4)^(3/ 4)*sqrt(-b*x^2 + a)*a^2*b^2*x*(-1/(a^3*b^2))^(3/4) + (-b*x^2 + a)^(1/4)*a^ 2*b*sqrt(-1/(a^3*b^2)) - I*(1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) + (-b*x^ 2 + a)^(3/4))/(b*x^2 - 2*a)) - 1/4*I*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log( (-2*I*(1/4)^(3/4)*sqrt(-b*x^2 + a)*a^2*b^2*x*(-1/(a^3*b^2))^(3/4) + (-b*x^ 2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) + I*(1/4)^(1/4)*a*b*x*(-1/(a^3*b^2)) ^(1/4) + (-b*x^2 + a)^(3/4))/(b*x^2 - 2*a))
\[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=- \int \frac {1}{- 2 a \sqrt [4]{a - b x^{2}} + b x^{2} \sqrt [4]{a - b x^{2}}}\, dx \]
\[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{1/4}\,\left (2\,a-b\,x^2\right )} \,d x \]