Integrand size = 24, antiderivative size = 90 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1}{a^4 x}-\frac {b x}{6 a^2 \left (a+b x^2\right )^3}-\frac {11 b x}{24 a^3 \left (a+b x^2\right )^2}-\frac {19 b x}{16 a^4 \left (a+b x^2\right )}-\frac {35 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{9/2}} \] Output:
-1/a^4/x-1/6*b*x/a^2/(b*x^2+a)^3-11/24*b*x/a^3/(b*x^2+a)^2-19/16*b*x/a^4/( b*x^2+a)-35/16*b^(1/2)*arctan(b^(1/2)*x/a^(1/2))/a^(9/2)
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {48 a^3+231 a^2 b x^2+280 a b^2 x^4+105 b^3 x^6}{48 a^4 x \left (a+b x^2\right )^3}-\frac {35 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{9/2}} \] Input:
Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
Output:
-1/48*(48*a^3 + 231*a^2*b*x^2 + 280*a*b^2*x^4 + 105*b^3*x^6)/(a^4*x*(a + b *x^2)^3) - (35*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(9/2))
Time = 0.37 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1380, 27, 253, 253, 253, 264, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^4 \int \frac {1}{b^4 x^2 \left (b x^2+a\right )^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^4}dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \int \frac {1}{x^2 \left (b x^2+a\right )^3}dx}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \left (\frac {5 \int \frac {1}{x^2 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{2 a}+\frac {1}{2 a x \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{2 a}+\frac {1}{2 a x \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\) |
Input:
Int[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
Output:
1/(6*a*x*(a + b*x^2)^3) + (7*(1/(4*a*x*(a + b*x^2)^2) + (5*(1/(2*a*x*(a + b*x^2)) + (3*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))) /(2*a)))/(4*a)))/(6*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {b \left (\frac {\frac {19}{16} x^{5} b^{2}+\frac {17}{6} a b \,x^{3}+\frac {29}{16} a^{2} x}{\left (b \,x^{2}+a \right )^{3}}+\frac {35 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{a^{4}}-\frac {1}{a^{4} x}\) | \(65\) |
risch | \(\frac {-\frac {35 b^{3} x^{6}}{16 a^{4}}-\frac {35 b^{2} x^{4}}{6 a^{3}}-\frac {77 b \,x^{2}}{16 a^{2}}-\frac {1}{a}}{x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (b \,x^{2}+a \right )}+\frac {35 \sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{32 a^{5}}-\frac {35 \sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{32 a^{5}}\) | \(120\) |
Input:
int(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
-b/a^4*((19/16*x^5*b^2+17/6*a*b*x^3+29/16*a^2*x)/(b*x^2+a)^3+35/16/(a*b)^( 1/2)*arctan(b/(a*b)^(1/2)*x))-1/a^4/x
Time = 0.08 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.98 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\left [-\frac {210 \, b^{3} x^{6} + 560 \, a b^{2} x^{4} + 462 \, a^{2} b x^{2} + 96 \, a^{3} - 105 \, {\left (b^{3} x^{7} + 3 \, a b^{2} x^{5} + 3 \, a^{2} b x^{3} + a^{3} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{96 \, {\left (a^{4} b^{3} x^{7} + 3 \, a^{5} b^{2} x^{5} + 3 \, a^{6} b x^{3} + a^{7} x\right )}}, -\frac {105 \, b^{3} x^{6} + 280 \, a b^{2} x^{4} + 231 \, a^{2} b x^{2} + 48 \, a^{3} + 105 \, {\left (b^{3} x^{7} + 3 \, a b^{2} x^{5} + 3 \, a^{2} b x^{3} + a^{3} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{48 \, {\left (a^{4} b^{3} x^{7} + 3 \, a^{5} b^{2} x^{5} + 3 \, a^{6} b x^{3} + a^{7} x\right )}}\right ] \] Input:
integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
[-1/96*(210*b^3*x^6 + 560*a*b^2*x^4 + 462*a^2*b*x^2 + 96*a^3 - 105*(b^3*x^ 7 + 3*a*b^2*x^5 + 3*a^2*b*x^3 + a^3*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt( -b/a) - a)/(b*x^2 + a)))/(a^4*b^3*x^7 + 3*a^5*b^2*x^5 + 3*a^6*b*x^3 + a^7* x), -1/48*(105*b^3*x^6 + 280*a*b^2*x^4 + 231*a^2*b*x^2 + 48*a^3 + 105*(b^3 *x^7 + 3*a*b^2*x^5 + 3*a^2*b*x^3 + a^3*x)*sqrt(b/a)*arctan(x*sqrt(b/a)))/( a^4*b^3*x^7 + 3*a^5*b^2*x^5 + 3*a^6*b*x^3 + a^7*x)]
Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {35 \sqrt {- \frac {b}{a^{9}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b}{a^{9}}}}{b} + x \right )}}{32} - \frac {35 \sqrt {- \frac {b}{a^{9}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b}{a^{9}}}}{b} + x \right )}}{32} + \frac {- 48 a^{3} - 231 a^{2} b x^{2} - 280 a b^{2} x^{4} - 105 b^{3} x^{6}}{48 a^{7} x + 144 a^{6} b x^{3} + 144 a^{5} b^{2} x^{5} + 48 a^{4} b^{3} x^{7}} \] Input:
integrate(1/x**2/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
35*sqrt(-b/a**9)*log(-a**5*sqrt(-b/a**9)/b + x)/32 - 35*sqrt(-b/a**9)*log( a**5*sqrt(-b/a**9)/b + x)/32 + (-48*a**3 - 231*a**2*b*x**2 - 280*a*b**2*x* *4 - 105*b**3*x**6)/(48*a**7*x + 144*a**6*b*x**3 + 144*a**5*b**2*x**5 + 48 *a**4*b**3*x**7)
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {105 \, b^{3} x^{6} + 280 \, a b^{2} x^{4} + 231 \, a^{2} b x^{2} + 48 \, a^{3}}{48 \, {\left (a^{4} b^{3} x^{7} + 3 \, a^{5} b^{2} x^{5} + 3 \, a^{6} b x^{3} + a^{7} x\right )}} - \frac {35 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{4}} \] Input:
integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
-1/48*(105*b^3*x^6 + 280*a*b^2*x^4 + 231*a^2*b*x^2 + 48*a^3)/(a^4*b^3*x^7 + 3*a^5*b^2*x^5 + 3*a^6*b*x^3 + a^7*x) - 35/16*b*arctan(b*x/sqrt(a*b))/(sq rt(a*b)*a^4)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {35 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{4}} - \frac {1}{a^{4} x} - \frac {57 \, b^{3} x^{5} + 136 \, a b^{2} x^{3} + 87 \, a^{2} b x}{48 \, {\left (b x^{2} + a\right )}^{3} a^{4}} \] Input:
integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
-35/16*b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/(a^4*x) - 1/48*(57*b^3* x^5 + 136*a*b^2*x^3 + 87*a^2*b*x)/((b*x^2 + a)^3*a^4)
Time = 17.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {1}{a}+\frac {77\,b\,x^2}{16\,a^2}+\frac {35\,b^2\,x^4}{6\,a^3}+\frac {35\,b^3\,x^6}{16\,a^4}}{a^3\,x+3\,a^2\,b\,x^3+3\,a\,b^2\,x^5+b^3\,x^7}-\frac {35\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,a^{9/2}} \] Input:
int(1/(x^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^2),x)
Output:
- (1/a + (77*b*x^2)/(16*a^2) + (35*b^2*x^4)/(6*a^3) + (35*b^3*x^6)/(16*a^4 ))/(a^3*x + b^3*x^7 + 3*a^2*b*x^3 + 3*a*b^2*x^5) - (35*b^(1/2)*atan((b^(1/ 2)*x)/a^(1/2)))/(16*a^(9/2))
Time = 0.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} x -315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,x^{3}-315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} x^{5}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{7}-48 a^{4}-231 a^{3} b \,x^{2}-280 a^{2} b^{2} x^{4}-105 a \,b^{3} x^{6}}{48 a^{5} x \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:
int(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
( - 105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*x - 315*sqrt(b) *sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*x**3 - 315*sqrt(b)*sqrt(a)*a tan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*x**5 - 105*sqrt(b)*sqrt(a)*atan((b*x)/ (sqrt(b)*sqrt(a)))*b**3*x**7 - 48*a**4 - 231*a**3*b*x**2 - 280*a**2*b**2*x **4 - 105*a*b**3*x**6)/(48*a**5*x*(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6))