Integrand size = 27, antiderivative size = 72 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{4 a^2 x^4}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a}\right )}{4 a^3} \] Output:
-1/4*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a^2/x^4-1/4*b^2*arctanh((-b*x^2+a)^( 1/2)*(b*x^2+a)^(1/2)/a)/a^3
Time = 0.35 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {a \sqrt {a-b x^2} \sqrt {a+b x^2}+2 b^2 x^4 \text {arctanh}\left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right )}{4 a^3 x^4} \] Input:
Integrate[1/(x^5*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]),x]
Output:
-1/4*(a*Sqrt[a - b*x^2]*Sqrt[a + b*x^2] + 2*b^2*x^4*ArcTanh[Sqrt[a - b*x^2 ]/Sqrt[a + b*x^2]])/(a^3*x^4)
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {338, 114, 25, 27, 103, 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.
\(\displaystyle \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx\) |
\(\Big \downarrow \) 338 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {b^2}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {b^2}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {b^2 \int \frac {1}{x^2 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {1}{2} \left (-\frac {b^3 \int \frac {1}{a^2 b-b x^4}d\left (\sqrt {a-b x^2} \sqrt {b x^2+a}\right )}{2 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{a}\right )}{2 a^3}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{2 a^2 x^4}\right )\) |
Input:
Int[1/(x^5*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]),x]
Output:
(-1/2*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/(a^2*x^4) - (b^2*ArcTanh[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/a])/(2*a^3))/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_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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_)^(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 = 0.57 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33
method | result | size |
default | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (\ln \left (\frac {2 a \left (\operatorname {csgn}\left (a \right ) \sqrt {-b^{2} x^{4}+a^{2}}+a \right )}{x^{2}}\right ) b^{2} x^{4}+\operatorname {csgn}\left (a \right ) a \sqrt {-b^{2} x^{4}+a^{2}}\right ) \operatorname {csgn}\left (a \right )}{4 a^{3} \sqrt {-b^{2} x^{4}+a^{2}}\, x^{4}}\) | \(96\) |
elliptic | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {\sqrt {-b^{2} x^{4}+a^{2}}}{4 a^{2} x^{4}}-\frac {b^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {-b^{2} x^{4}+a^{2}}}{x^{2}}\right )}{4 a^{2} \sqrt {a^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(103\) |
risch | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{4 a^{2} x^{4}}-\frac {b^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {-b^{2} x^{4}+a^{2}}}{x^{2}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{4 a^{2} \sqrt {a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\) | \(111\) |
Input:
int(1/x^5/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a^3*(ln(2*a*(csgn(a)*(-b^2*x^4+a^2)^ (1/2)+a)/x^2)*b^2*x^4+csgn(a)*a*(-b^2*x^4+a^2)^(1/2))*csgn(a)/(-b^2*x^4+a^ 2)^(1/2)/x^4
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {b^{2} x^{4} \log \left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} + a\right ) - b^{2} x^{4} \log \left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} - a\right ) + 2 \, \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} a}{8 \, a^{3} x^{4}} \] Input:
integrate(1/x^5/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-1/8*(b^2*x^4*log(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a) + a) - b^2*x^4*log(sqrt (b*x^2 + a)*sqrt(-b*x^2 + a) - a) + 2*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)*a)/ (a^3*x^4)
Timed out. \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:
integrate(1/x**5/(-b*x**2+a)**(1/2)/(b*x**2+a)**(1/2),x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {b^{2} \log \left (\frac {2 \, a^{2}}{x^{2}} + \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} a}{x^{2}}\right )}{4 \, a^{3}} - \frac {\sqrt {-b^{2} x^{4} + a^{2}}}{4 \, a^{2} x^{4}} \] Input:
integrate(1/x^5/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
-1/4*b^2*log(2*a^2/x^2 + 2*sqrt(-b^2*x^4 + a^2)*a/x^2)/a^3 - 1/4*sqrt(-b^2 *x^4 + a^2)/(a^2*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (60) = 120\).
Time = 0.22 (sec) , antiderivative size = 370, normalized size of antiderivative = 5.14 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {\frac {b^{3} \log \left ({\left | -\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} + \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}} + 2 \right |}\right )}{a^{3}} - \frac {b^{3} \log \left ({\left | -\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} + \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}} - 2 \right |}\right )}{a^{3}} - \frac {4 \, {\left (b^{3} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{3} + 4 \, b^{3} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}{\sqrt {b x^{2} + a}} - \frac {\sqrt {b x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-b x^{2} + a}}\right )}^{2} - 4\right )}^{2} a^{3}}}{4 \, b} \] Input:
integrate(1/x^5/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
-1/4*(b^3*log(abs(-(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))/a^3 - b^3*log(a bs(-(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))/a^3 - 4*(b^3*((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)))^3 + 4*b^3*((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*a^3))/b
Time = 5.89 (sec) , antiderivative size = 344, normalized size of antiderivative = 4.78 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\frac {b^2\,\ln \left (\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-1\right )}{4\,a^3}-\frac {\frac {b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-\frac {b^2}{2}+\frac {15\,b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{2\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4}}{\frac {32\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}-\frac {64\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^4}+\frac {32\,a^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^6}}-\frac {b^2\,\ln \left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {a-b\,x^2}-\sqrt {a}}\right )}{4\,a^3}+\frac {b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{64\,a^3\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2} \] Input:
int(1/(x^5*(a + b*x^2)^(1/2)*(a - b*x^2)^(1/2)),x)
Output:
(b^2*log(((a + b*x^2)^(1/2) - a^(1/2))^2/((a - b*x^2)^(1/2) - a^(1/2))^2 - 1))/(4*a^3) - ((b^2*((a + b*x^2)^(1/2) - a^(1/2))^2)/((a - b*x^2)^(1/2) - a^(1/2))^2 - b^2/2 + (15*b^2*((a + b*x^2)^(1/2) - a^(1/2))^4)/(2*((a - b* x^2)^(1/2) - a^(1/2))^4))/((32*a^3*((a + b*x^2)^(1/2) - a^(1/2))^2)/((a - b*x^2)^(1/2) - a^(1/2))^2 - (64*a^3*((a + b*x^2)^(1/2) - a^(1/2))^4)/((a - b*x^2)^(1/2) - a^(1/2))^4 + (32*a^3*((a + b*x^2)^(1/2) - a^(1/2))^6)/((a - b*x^2)^(1/2) - a^(1/2))^6) - (b^2*log(((a + b*x^2)^(1/2) - a^(1/2))/((a - b*x^2)^(1/2) - a^(1/2))))/(4*a^3) + (b^2*((a + b*x^2)^(1/2) - a^(1/2))^2 )/(64*a^3*((a - b*x^2)^(1/2) - a^(1/2))^2)
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a +2 \,\mathrm {log}\left (\sqrt {-b \,x^{2}+a}-\sqrt {b \,x^{2}+a}\right ) b^{2} x^{4}-2 \,\mathrm {log}\left (x \right ) b^{2} x^{4}}{4 a^{3} x^{4}} \] Input:
int(1/x^5/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x)
Output:
( - sqrt(a + b*x**2)*sqrt(a - b*x**2)*a + 2*log(sqrt(a - b*x**2) - sqrt(a + b*x**2))*b**2*x**4 - 2*log(x)*b**2*x**4)/(4*a**3*x**4)