Integrand size = 29, antiderivative size = 124 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {x \left (a+b x^2\right )}{4 a^2 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a-b x^2} \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
1/4*x*(b*x^2+a)/a^2/(-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2)+3/8*arctanh(x*2^ (1/2)*b^(1/2)/(b*x^2+a)^(1/2))*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a^2*2^(1/2 )/b^(1/2)/(-b^2*x^4+a^2)^(1/2)
Time = 2.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {\sqrt {a^2-b^2 x^4} \left (2 \sqrt {b} x \sqrt {a+b x^2}+3 \sqrt {2} \left (a-b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )\right )}{8 a^2 \sqrt {b} \left (a-b x^2\right )^{3/2} \sqrt {a+b x^2}} \]
(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a + b*x^2] + 3*Sqrt[2]*(a - b*x^2)* ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a + b*x^2]]))/(8*a^2*Sqrt[b]*(a - b*x^2)^ (3/2)*Sqrt[a + b*x^2])
Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1396, 296, 291, 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.
| \(\displaystyle \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \int \frac {1}{\left (a-b x^2\right )^2 \sqrt {b x^2+a}}dx}{\sqrt {a^2-b^2 x^4}}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \left (\frac {3 \int \frac {1}{\left (a-b x^2\right ) \sqrt {b x^2+a}}dx}{4 a}+\frac {x \sqrt {a+b x^2}}{4 a^2 \left (a-b x^2\right )}\right )}{\sqrt {a^2-b^2 x^4}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \left (\frac {3 \int \frac {1}{a-\frac {2 a b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{4 a}+\frac {x \sqrt {a+b x^2}}{4 a^2 \left (a-b x^2\right )}\right )}{\sqrt {a^2-b^2 x^4}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \left (\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b}}+\frac {x \sqrt {a+b x^2}}{4 a^2 \left (a-b x^2\right )}\right )}{\sqrt {a^2-b^2 x^4}}\) |
(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*((x*Sqrt[a + b*x^2])/(4*a^2*(a - b*x^2)) + (3*ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a + b*x^2]])/(4*Sqrt[2]*a^2*Sqrt[b]) ))/Sqrt[a^2 - b^2*x^4]
3.3.9.3.1 Defintions of rubi rules used
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[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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(100)=200\).
Time = 0.23 (sec) , antiderivative size = 499, normalized size of antiderivative = 4.02
| method | result | size |
| default | \(-\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, b^{\frac {5}{2}} \left (-3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}+x \sqrt {a b}+a \right )}{b x -\sqrt {a b}}\right ) b^{\frac {3}{2}} x^{2} \sqrt {a}+3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}-x \sqrt {a b}+a \right )}{b x +\sqrt {a b}}\right ) b^{\frac {3}{2}} x^{2} \sqrt {a}+3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}+x \sqrt {a b}+a \right )}{b x -\sqrt {a b}}\right ) a^{\frac {3}{2}} \sqrt {b}-3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}-x \sqrt {a b}+a \right )}{b x +\sqrt {a b}}\right ) a^{\frac {3}{2}} \sqrt {b}+4 \ln \left (\frac {\sqrt {b}\, \sqrt {b \,x^{2}+a}+b x}{\sqrt {b}}\right ) b \,x^{2} \sqrt {a b}-4 \ln \left (\frac {\sqrt {b}\, \sqrt {-\frac {\left (-b x +\sqrt {-a b}\right ) \left (b x +\sqrt {-a b}\right )}{b}}+b x}{\sqrt {b}}\right ) b \,x^{2} \sqrt {a b}+4 \sqrt {b \,x^{2}+a}\, \sqrt {b}\, \sqrt {a b}\, x -4 \ln \left (\frac {\sqrt {b}\, \sqrt {b \,x^{2}+a}+b x}{\sqrt {b}}\right ) a \sqrt {a b}+4 \ln \left (\frac {\sqrt {b}\, \sqrt {-\frac {\left (-b x +\sqrt {-a b}\right ) \left (b x +\sqrt {-a b}\right )}{b}}+b x}{\sqrt {b}}\right ) a \sqrt {a b}\right )}{4 \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (b x -\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right ) \sqrt {a b}}\) | \(499\) |
-1/4*(-b^2*x^4+a^2)^(1/2)*b^(5/2)*(-3*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(b*x ^2+a)^(1/2)+x*(a*b)^(1/2)+a)/(b*x-(a*b)^(1/2)))*b^(3/2)*x^2*a^(1/2)+3*2^(1 /2)*ln(2*b*(2^(1/2)*a^(1/2)*(b*x^2+a)^(1/2)-x*(a*b)^(1/2)+a)/(b*x+(a*b)^(1 /2)))*b^(3/2)*x^2*a^(1/2)+3*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(b*x^2+a)^(1/2 )+x*(a*b)^(1/2)+a)/(b*x-(a*b)^(1/2)))*a^(3/2)*b^(1/2)-3*2^(1/2)*ln(2*b*(2^ (1/2)*a^(1/2)*(b*x^2+a)^(1/2)-x*(a*b)^(1/2)+a)/(b*x+(a*b)^(1/2)))*a^(3/2)* b^(1/2)+4*ln((b^(1/2)*(b*x^2+a)^(1/2)+b*x)/b^(1/2))*b*x^2*(a*b)^(1/2)-4*ln ((b^(1/2)*(-1/b*(-b*x+(-a*b)^(1/2))*(b*x+(-a*b)^(1/2)))^(1/2)+b*x)/b^(1/2) )*b*x^2*(a*b)^(1/2)+4*(b*x^2+a)^(1/2)*b^(1/2)*(a*b)^(1/2)*x-4*ln((b^(1/2)* (b*x^2+a)^(1/2)+b*x)/b^(1/2))*a*(a*b)^(1/2)+4*ln((b^(1/2)*(-1/b*(-b*x+(-a* b)^(1/2))*(b*x+(-a*b)^(1/2)))^(1/2)+b*x)/b^(1/2))*a*(a*b)^(1/2))/(-b*x^2+a )^(1/2)/(b*x^2+a)^(1/2)/(-(-a*b)^(1/2)+(a*b)^(1/2))^2/((-a*b)^(1/2)+(a*b)^ (1/2))^2/(b*x-(a*b)^(1/2))/(b*x+(a*b)^(1/2))/(a*b)^(1/2)
Time = 0.25 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.44 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\left [\frac {4 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} b x + 3 \, \sqrt {2} {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt {b} \log \left (-\frac {3 \, b^{2} x^{4} - 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {b} x - a^{2}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}\right )}{16 \, {\left (a^{2} b^{3} x^{4} - 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}, \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} b x + 3 \, \sqrt {2} {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {-b}}{2 \, {\left (b^{2} x^{3} - a b x\right )}}\right )}{8 \, {\left (a^{2} b^{3} x^{4} - 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}\right ] \]
[1/16*(4*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*b*x + 3*sqrt(2)*(b^2*x^4 - 2*a*b*x^2 + a^2)*sqrt(b)*log(-(3*b^2*x^4 - 2*a*b*x^2 - 2*sqrt(2)*sqrt(-b^2 *x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(b)*x - a^2)/(b^2*x^4 - 2*a*b*x^2 + a^2)) )/(a^2*b^3*x^4 - 2*a^3*b^2*x^2 + a^4*b), 1/8*(2*sqrt(-b^2*x^4 + a^2)*sqrt( -b*x^2 + a)*b*x + 3*sqrt(2)*(b^2*x^4 - 2*a*b*x^2 + a^2)*sqrt(-b)*arctan(1/ 2*sqrt(2)*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(-b)/(b^2*x^3 - a*b*x) ))/(a^2*b^3*x^4 - 2*a^3*b^2*x^2 + a^4*b)]
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {a^2-b^2\,x^4}\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \]