Integrand size = 27, antiderivative size = 59 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 x^2}-b \arctan \left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b x^2}}\right ) \] Output:
-1/2*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^2-b*arctan((b*x^2+a)^(1/2)/(-b*x^2 +a)^(1/2))
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 x^2}+b \arctan \left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right ) \] Input:
Integrate[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^3,x]
Output:
-1/2*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^2 + b*ArcTan[Sqrt[a - b*x^2]/Sqrt [a + b*x^2]]
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {338, 108, 25, 27, 45, 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.
\(\displaystyle \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx\) |
\(\Big \downarrow \) 338 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {a-b x^2} \sqrt {b x^2+a}}{x^4}dx^2\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{2} \left (\int -\frac {b^2}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {b^2}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (b^2 \left (-\int \frac {1}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {1}{2} \left (-2 b^2 \int \frac {1}{-b x^4-b}d\frac {\sqrt {a-b x^2}}{\sqrt {b x^2+a}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (2 b \arctan \left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^2}\right )\) |
Input:
Int[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^3,x]
Output:
(-((Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^2) + 2*b*ArcTan[Sqrt[a - b*x^2]/Sqr t[a + b*x^2]])/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && IntegerQ[(m - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.58 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.46
method | result | size |
default | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (b \right ) b \,x^{2}}{\sqrt {-b^{2} x^{4}+a^{2}}}\right ) b \,x^{2}+\operatorname {csgn}\left (b \right ) \sqrt {-b^{2} x^{4}+a^{2}}\right ) \operatorname {csgn}\left (b \right )}{2 \sqrt {-b^{2} x^{4}+a^{2}}\, x^{2}}\) | \(86\) |
risch | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{2 x^{2}}-\frac {b^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x^{2}}{\sqrt {-b^{2} x^{4}+a^{2}}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{2 \sqrt {b^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(97\) |
elliptic | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {\left (-b^{2} x^{4}+a^{2}\right )^{\frac {3}{2}}}{2 a^{2} x^{2}}-\frac {b^{2} x^{2} \sqrt {-b^{2} x^{4}+a^{2}}}{2 a^{2}}-\frac {b^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x^{2}}{\sqrt {-b^{2} x^{4}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(117\) |
Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)*(arctan(csgn(b)*b*x^2/(-b^2*x^4+a^2) ^(1/2))*b*x^2+csgn(b)*(-b^2*x^4+a^2)^(1/2))*csgn(b)/(-b^2*x^4+a^2)^(1/2)/x ^2
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=\frac {2 \, b x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} - a}{b x^{2}}\right ) - \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{2 \, x^{2}} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^3,x, algorithm="fricas")
Output:
1/2*(2*b*x^2*arctan((sqrt(b*x^2 + a)*sqrt(-b*x^2 + a) - a)/(b*x^2)) - sqrt (b*x^2 + a)*sqrt(-b*x^2 + a))/x^2
\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=\int \frac {\sqrt {a - b x^{2}} \sqrt {a + b x^{2}}}{x^{3}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2)/x**3,x)
Output:
Integral(sqrt(a - b*x**2)*sqrt(a + b*x**2)/x**3, x)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=-\frac {1}{2} \, b \arcsin \left (\frac {b x^{2}}{a}\right ) - \frac {\sqrt {-b^{2} x^{4} + a^{2}}}{2 \, x^{2}} \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^3,x, algorithm="maxima")
Output:
-1/2*b*arcsin(b*x^2/a) - 1/2*sqrt(-b^2*x^4 + a^2)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (49) = 98\).
Time = 0.20 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=-\frac {1}{2} \, {\left (\pi + \frac {4 \, {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{2} - 4} + 2 \, \arctan \left (\frac {\sqrt {b x^{2} + a} {\left (\frac {{\left (\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}\right )}^{2}}{b x^{2} + a} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}\right )}}\right )\right )} b \] Input:
integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^3,x, algorithm="giac")
Output:
-1/2*(pi + 4*((sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))/sqrt(b*x^2 + a) - sqrt( b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a)))/(((sqrt(2)*sqrt(a) - sqrt (-b*x^2 + a))/sqrt(b*x^2 + a) - sqrt(b*x^2 + a)/(sqrt(2)*sqrt(a) - sqrt(-b *x^2 + a)))^2 - 4) + 2*arctan(1/2*sqrt(b*x^2 + a)*((sqrt(2)*sqrt(a) - sqrt (-b*x^2 + a))^2/(b*x^2 + a) - 1)/(sqrt(2)*sqrt(a) - sqrt(-b*x^2 + a))))*b
Time = 2.95 (sec) , antiderivative size = 193, normalized size of antiderivative = 3.27 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=\frac {b\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{8\,\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}-\frac {b-\frac {5\,b\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}}{\frac {8\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {a-b\,x^2}-\sqrt {a}}-\frac {8\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^3}}-2\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {a-b\,x^2}-\sqrt {a}}\right ) \] Input:
int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^3,x)
Output:
(b*((a + b*x^2)^(1/2) - a^(1/2)))/(8*((a - b*x^2)^(1/2) - a^(1/2))) - (b - (5*b*((a + b*x^2)^(1/2) - a^(1/2))^2)/((a - b*x^2)^(1/2) - a^(1/2))^2)/(( 8*((a + b*x^2)^(1/2) - a^(1/2)))/((a - b*x^2)^(1/2) - a^(1/2)) - (8*((a + b*x^2)^(1/2) - a^(1/2))^3)/((a - b*x^2)^(1/2) - a^(1/2))^3) - 2*b*atan(((a + b*x^2)^(1/2) - a^(1/2))/((a - b*x^2)^(1/2) - a^(1/2)))
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^3} \, dx=\frac {-2 \mathit {atan} \left (\frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{2}+a}\right ) b \,x^{2}-\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{2 x^{2}} \] Input:
int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^3,x)
Output:
( - 2*atan((sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a - b*x**2))*b*x**2 - sqrt (a + b*x**2)*sqrt(a - b*x**2))/(2*x**2)