Integrand size = 38, antiderivative size = 48 \[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}\right )}{\sqrt {2} \sqrt {b}} \] Output:
1/2*arctan(2^(1/2)*b^(1/2)*x/(-b*x^2+(b^2*x^4+a)^(1/2))^(1/2))*2^(1/2)/b^( 1/2)
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\frac {i \log \left (-i b^{3/2} x^2+i \sqrt {b} \sqrt {a+b^2 x^4}+\sqrt {2} b x \sqrt {-b x^2+\sqrt {a+b^2 x^4}}\right )}{\sqrt {2} \sqrt {b}} \] Input:
Integrate[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]
Output:
(I*Log[(-I)*b^(3/2)*x^2 + I*Sqrt[b]*Sqrt[a + b^2*x^4] + Sqrt[2]*b*x*Sqrt[- (b*x^2) + Sqrt[a + b^2*x^4]]])/(Sqrt[2]*Sqrt[b])
Time = 0.41 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2557, 216}
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {a+b^2 x^4}-b x^2}}{\sqrt {a+b^2 x^4}} \, dx\) |
\(\Big \downarrow \) 2557 |
\(\displaystyle \int \frac {1}{\frac {2 b x^2}{\sqrt {a+b^2 x^4}-b x^2}+1}d\frac {x}{\sqrt {\sqrt {a+b^2 x^4}-b x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\sqrt {a+b^2 x^4}-b x^2}}\right )}{\sqrt {2} \sqrt {b}}\) |
Input:
Int[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]
Output:
ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sq rt[b])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)* (x_)^4], x_Symbol] :> Simp[d Subst[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x ^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d^2, 0]
\[\int \frac {\sqrt {-b \,x^{2}+\sqrt {b^{2} x^{4}+a}}}{\sqrt {b^{2} x^{4}+a}}d x\]
Input:
int((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)
Output:
int((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)
Time = 0.90 (sec) , antiderivative size = 169, normalized size of antiderivative = 3.52 \[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{b}} \log \left (4 \, b^{2} x^{4} - 4 \, \sqrt {b^{2} x^{4} + a} b x^{2} + 2 \, {\left (\sqrt {2} b^{2} x^{3} \sqrt {-\frac {1}{b}} - \sqrt {2} \sqrt {b^{2} x^{4} + a} b x \sqrt {-\frac {1}{b}}\right )} \sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}} + a\right ), \frac {\sqrt {2} \arctan \left (\frac {{\left (\sqrt {2} b^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {b^{2} x^{4} + a} \sqrt {b} x\right )} \sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{a}\right )}{2 \, \sqrt {b}}\right ] \] Input:
integrate((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x, algorithm= "fricas")
Output:
[1/4*sqrt(2)*sqrt(-1/b)*log(4*b^2*x^4 - 4*sqrt(b^2*x^4 + a)*b*x^2 + 2*(sqr t(2)*b^2*x^3*sqrt(-1/b) - sqrt(2)*sqrt(b^2*x^4 + a)*b*x*sqrt(-1/b))*sqrt(- b*x^2 + sqrt(b^2*x^4 + a)) + a), 1/2*sqrt(2)*arctan((sqrt(2)*b^(3/2)*x^3 + sqrt(2)*sqrt(b^2*x^4 + a)*sqrt(b)*x)*sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/a)/ sqrt(b)]
\[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\int \frac {\sqrt {- b x^{2} + \sqrt {a + b^{2} x^{4}}}}{\sqrt {a + b^{2} x^{4}}}\, dx \] Input:
integrate((-b*x**2+(b**2*x**4+a)**(1/2))**(1/2)/(b**2*x**4+a)**(1/2),x)
Output:
Integral(sqrt(-b*x**2 + sqrt(a + b**2*x**4))/sqrt(a + b**2*x**4), x)
\[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\int { \frac {\sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{\sqrt {b^{2} x^{4} + a}} \,d x } \] Input:
integrate((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x, algorithm= "maxima")
Output:
integrate(sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)
\[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\int { \frac {\sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{\sqrt {b^{2} x^{4} + a}} \,d x } \] Input:
integrate((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x, algorithm= "giac")
Output:
integrate(sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)
Timed out. \[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\int \frac {\sqrt {\sqrt {b^2\,x^4+a}-b\,x^2}}{\sqrt {b^2\,x^4+a}} \,d x \] Input:
int(((a + b^2*x^4)^(1/2) - b*x^2)^(1/2)/(a + b^2*x^4)^(1/2),x)
Output:
int(((a + b^2*x^4)^(1/2) - b*x^2)^(1/2)/(a + b^2*x^4)^(1/2), x)
\[ \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx=\int \frac {\sqrt {b^{2} x^{4}+a}\, \sqrt {\sqrt {b^{2} x^{4}+a}-b \,x^{2}}}{b^{2} x^{4}+a}d x \] Input:
int((-b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)
Output:
int((sqrt(a + b**2*x**4)*sqrt(sqrt(a + b**2*x**4) - b*x**2))/(a + b**2*x** 4),x)