Integrand size = 23, antiderivative size = 70 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {b+a x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {d} \sqrt {a+\frac {b}{x^2}} x} \] Output:
(a*x^2+b)^(1/2)*arctanh(d^(1/2)*(a*x^2+b)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a ^(1/2)/d^(1/2)/(a+b/x^2)^(1/2)/x
Time = 0.60 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {b+a x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {d} \sqrt {a+\frac {b}{x^2}} x} \] Input:
Integrate[1/(Sqrt[a + b/x^2]*Sqrt[c + d*x^2]),x]
Output:
(Sqrt[b + a*x^2]*ArcTanh[(Sqrt[d]*Sqrt[b + a*x^2])/(Sqrt[a]*Sqrt[c + d*x^2 ])])/(Sqrt[a]*Sqrt[d]*Sqrt[a + b/x^2]*x)
Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {942, 353, 66, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx\) |
\(\Big \downarrow \) 942 |
\(\displaystyle \frac {\sqrt {a x^2+b} \int \frac {x}{\sqrt {a x^2+b} \sqrt {d x^2+c}}dx}{x \sqrt {a+\frac {b}{x^2}}}\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {\sqrt {a x^2+b} \int \frac {1}{\sqrt {a x^2+b} \sqrt {d x^2+c}}dx^2}{2 x \sqrt {a+\frac {b}{x^2}}}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {\sqrt {a x^2+b} \int \frac {1}{a-d x^4}d\frac {\sqrt {a x^2+b}}{\sqrt {d x^2+c}}}{x \sqrt {a+\frac {b}{x^2}}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {a x^2+b} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a x^2+b}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {d} x \sqrt {a+\frac {b}{x^2}}}\) |
Input:
Int[1/(Sqrt[a + b/x^2]*Sqrt[c + d*x^2]),x]
Output:
(Sqrt[b + a*x^2]*ArcTanh[(Sqrt[d]*Sqrt[b + a*x^2])/(Sqrt[a]*Sqrt[c + d*x^2 ])])/(Sqrt[a]*Sqrt[d]*Sqrt[a + b/x^2]*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] && !GtQ[c - a*(d/b), 0]
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_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symb ol] :> Simp[x^(n*FracPart[q])*((c + d/x^n)^FracPart[q]/(d + c*x^n)^FracPart [q]) Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[mn, -n] && !IntegerQ[q] && !IntegerQ[p]
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {\left (a \,x^{2}+b \right ) \ln \left (\frac {2 a \,x^{2} d +2 \sqrt {\left (a \,x^{2}+b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a d}+a c +b d}{2 \sqrt {a d}}\right ) \sqrt {d \,x^{2}+c}}{2 \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {a d}\, \sqrt {\left (a \,x^{2}+b \right ) \left (d \,x^{2}+c \right )}}\) | \(103\) |
Input:
int(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)*ln(1/2*(2*a*x^2*d+2*((a*x^2+b)*(d*x^ 2+c))^(1/2)*(a*d)^(1/2)+a*c+b*d)/(a*d)^(1/2))*(d*x^2+c)^(1/2)/(a*d)^(1/2)/ ((a*x^2+b)*(d*x^2+c))^(1/2)
Time = 0.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.97 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\left [\frac {\sqrt {a d} \log \left (8 \, a^{2} d^{2} x^{4} + a^{2} c^{2} + 6 \, a b c d + b^{2} d^{2} + 8 \, {\left (a^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, a d x^{3} + {\left (a c + b d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {a d} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right )}{4 \, a d}, -\frac {\sqrt {-a d} \arctan \left (\frac {{\left (2 \, a d x^{3} + {\left (a c + b d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-a d} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{2 \, {\left (a^{2} d^{2} x^{4} + a b c d + {\left (a^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, a d}\right ] \] Input:
integrate(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
[1/4*sqrt(a*d)*log(8*a^2*d^2*x^4 + a^2*c^2 + 6*a*b*c*d + b^2*d^2 + 8*(a^2* c*d + a*b*d^2)*x^2 + 4*(2*a*d*x^3 + (a*c + b*d)*x)*sqrt(d*x^2 + c)*sqrt(a* d)*sqrt((a*x^2 + b)/x^2))/(a*d), -1/2*sqrt(-a*d)*arctan(1/2*(2*a*d*x^3 + ( a*c + b*d)*x)*sqrt(d*x^2 + c)*sqrt(-a*d)*sqrt((a*x^2 + b)/x^2)/(a^2*d^2*x^ 4 + a*b*c*d + (a^2*c*d + a*b*d^2)*x^2))/(a*d)]
\[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\sqrt {a + \frac {b}{x^{2}}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(1/(a+b/x**2)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(1/(sqrt(a + b/x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{\sqrt {d x^{2} + c} \sqrt {a + \frac {b}{x^{2}}}} \,d x } \] Input:
integrate(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(d*x^2 + c)*sqrt(a + b/x^2)), x)
Time = 0.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\frac {a \log \left ({\left | -\sqrt {a d} \sqrt {b} + \sqrt {a^{2} c} \right |}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {a d} {\left | a \right |}} - \frac {a \log \left ({\left | -\sqrt {a x^{2} + b} \sqrt {a d} + \sqrt {a^{2} c + {\left (a x^{2} + b\right )} a d - a b d} \right |}\right )}{\sqrt {a d} {\left | a \right |} \mathrm {sgn}\left (x\right )} \] Input:
integrate(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
a*log(abs(-sqrt(a*d)*sqrt(b) + sqrt(a^2*c)))*sgn(x)/(sqrt(a*d)*abs(a)) - a *log(abs(-sqrt(a*x^2 + b)*sqrt(a*d) + sqrt(a^2*c + (a*x^2 + b)*a*d - a*b*d )))/(sqrt(a*d)*abs(a)*sgn(x))
Timed out. \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{x^2}}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int(1/((a + b/x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int(1/((a + b/x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a \,x^{2}+b}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, a \right )}{a d} \] Input:
int(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(d)*sqrt(a)*log( - sqrt(a)*sqrt(a*x**2 + b)*d - sqrt(d)*sqrt(c + d*x* *2)*a))/(a*d)