Integrand size = 20, antiderivative size = 76 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=\frac {35}{16} \sqrt {-1+\frac {1}{x^2}}-\frac {35}{48} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6-\frac {35}{16} \arctan \left (\sqrt {-1+\frac {1}{x^2}}\right ) \]
-35/48*(-1+1/x^2)^(3/2)*x^2-7/24*(-1+1/x^2)^(5/2)*x^4-1/6*(-1+1/x^2)^(7/2) *x^6-35/16*arctan((-1+1/x^2)^(1/2))+35/16*(-1+1/x^2)^(1/2)
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=\frac {1}{48} \sqrt {-1+\frac {1}{x^2}} \left (48+87 x^2-38 x^4+8 x^6\right )-\frac {35 \sqrt {-1+\frac {1}{x^2}} x \text {arctanh}\left (\frac {\sqrt {-1+x^2}}{-1+x}\right )}{8 \sqrt {-1+x^2}} \]
(Sqrt[-1 + x^(-2)]*(48 + 87*x^2 - 38*x^4 + 8*x^6))/48 - (35*Sqrt[-1 + x^(- 2)]*x*ArcTanh[Sqrt[-1 + x^2]/(-1 + x)])/(8*Sqrt[-1 + x^2])
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1016, 281, 798, 51, 51, 51, 60, 73, 216}
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.
\(\displaystyle \int \frac {\sqrt {\frac {1}{x^2}-1} \left (x^2-1\right )^3}{x} \, dx\) |
\(\Big \downarrow \) 1016 |
\(\displaystyle \int \left (1-\frac {1}{x^2}\right )^3 \sqrt {\frac {1}{x^2}-1} x^5dx\) |
\(\Big \downarrow \) 281 |
\(\displaystyle -\int \left (\frac {1}{x^2}-1\right )^{7/2} x^5dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{2} \int \left (\frac {1}{x^2}-1\right )^{7/2} x^8d\frac {1}{x^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{6} \int \left (\frac {1}{x^2}-1\right )^{5/2} x^6d\frac {1}{x^2}-\frac {1}{3} \left (\frac {1}{x^2}-1\right )^{7/2} x^6\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{6} \left (\frac {5}{4} \int \left (\frac {1}{x^2}-1\right )^{3/2} x^4d\frac {1}{x^2}-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{5/2} x^4\right )-\frac {1}{3} \left (\frac {1}{x^2}-1\right )^{7/2} x^6\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \int \sqrt {\frac {1}{x^2}-1} x^2d\frac {1}{x^2}-\left (\frac {1}{x^2}-1\right )^{3/2} x^2\right )-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{5/2} x^4\right )-\frac {1}{3} \left (\frac {1}{x^2}-1\right )^{7/2} x^6\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \left (2 \sqrt {\frac {1}{x^2}-1}-\int \frac {x^2}{\sqrt {\frac {1}{x^2}-1}}d\frac {1}{x^2}\right )-\left (\frac {1}{x^2}-1\right )^{3/2} x^2\right )-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{5/2} x^4\right )-\frac {1}{3} \left (\frac {1}{x^2}-1\right )^{7/2} x^6\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \left (2 \sqrt {\frac {1}{x^2}-1}-2 \int \frac {1}{1+\frac {1}{x^4}}d\sqrt {\frac {1}{x^2}-1}\right )-\left (\frac {1}{x^2}-1\right )^{3/2} x^2\right )-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{5/2} x^4\right )-\frac {1}{3} \left (\frac {1}{x^2}-1\right )^{7/2} x^6\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{6} \left (\frac {5}{4} \left (\frac {3}{2} \left (2 \sqrt {\frac {1}{x^2}-1}-2 \arctan \left (\sqrt {\frac {1}{x^2}-1}\right )\right )-\left (\frac {1}{x^2}-1\right )^{3/2} x^2\right )-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{5/2} x^4\right )-\frac {1}{3} \left (\frac {1}{x^2}-1\right )^{7/2} x^6\right )\) |
(-1/3*((-1 + x^(-2))^(7/2)*x^6) + (7*(-1/2*((-1 + x^(-2))^(5/2)*x^4) + (5* (-((-1 + x^(-2))^(3/2)*x^2) + (3*(2*Sqrt[-1 + x^(-2)] - 2*ArcTan[Sqrt[-1 + x^(-2)]]))/2))/4))/6)/2
3.7.75.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^( p_.), x_Symbol] :> Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ [{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !I ntegerQ[p])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83
method | result | size |
trager | \(2 \left (\frac {1}{12} x^{6}-\frac {19}{48} x^{4}+\frac {29}{32} x^{2}+\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}-\frac {35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\left (\sqrt {-\frac {x^{2}-1}{x^{2}}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x \right )}{16}\) | \(63\) |
risch | \(\frac {\left (8 x^{8}-46 x^{6}+125 x^{4}-39 x^{2}-48\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{48 x^{2}-48}-\frac {35 \arcsin \left (x \right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}\, x \sqrt {-x^{2}+1}}{16 \left (x^{2}-1\right )}\) | \(78\) |
default | \(\frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (-8 x^{4} \left (-x^{2}+1\right )^{\frac {3}{2}}+30 x^{2} \left (-x^{2}+1\right )^{\frac {3}{2}}+48 \left (-x^{2}+1\right )^{\frac {3}{2}}+105 x^{2} \sqrt {-x^{2}+1}+105 \arcsin \left (x \right ) x \right )}{48 \sqrt {-x^{2}+1}}\) | \(83\) |
2*(1/12*x^6-19/48*x^4+29/32*x^2+1/2)*(-(x^2-1)/x^2)^(1/2)-35/16*RootOf(_Z^ 2+1)*ln(((-(x^2-1)/x^2)^(1/2)+RootOf(_Z^2+1))*x)
Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=\frac {1}{48} \, {\left (8 \, x^{6} - 38 \, x^{4} + 87 \, x^{2} + 48\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} - \frac {35}{8} \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]
1/48*(8*x^6 - 38*x^4 + 87*x^2 + 48)*sqrt(-(x^2 - 1)/x^2) - 35/8*arctan((x* sqrt(-(x^2 - 1)/x^2) - 1)/x)
Time = 51.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=- \frac {x^{6} \left (-1 + \frac {1}{x^{2}}\right )^{\frac {3}{2}}}{6} - \frac {5 x^{4} \sqrt {-1 + \frac {1}{x^{2}}} \cdot \left (2 - \frac {1}{x^{2}}\right )}{16} + \frac {3 x^{2} \sqrt {-1 + \frac {1}{x^{2}}}}{2} + \sqrt {-1 + \frac {1}{x^{2}}} - \frac {35 \operatorname {atan}{\left (\sqrt {-1 + \frac {1}{x^{2}}} \right )}}{16} \]
-x**6*(-1 + x**(-2))**(3/2)/6 - 5*x**4*sqrt(-1 + x**(-2))*(2 - 1/x**2)/16 + 3*x**2*sqrt(-1 + x**(-2))/2 + sqrt(-1 + x**(-2)) - 35*atan(sqrt(-1 + x** (-2)))/16
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (56) = 112\).
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=\frac {3}{2} \, x^{2} \sqrt {\frac {1}{x^{2}} - 1} + \sqrt {\frac {1}{x^{2}} - 1} - \frac {3 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\frac {1}{x^{2}} - 1}}{48 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{3} + 3 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {3}{x^{2}} - 2\right )}} + \frac {3 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {1}{x^{2}} - 1}\right )}}{8 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {2}{x^{2}} - 1\right )}} - \frac {35}{16} \, \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \]
3/2*x^2*sqrt(1/x^2 - 1) + sqrt(1/x^2 - 1) - 1/48*(3*(1/x^2 - 1)^(5/2) + 8* (1/x^2 - 1)^(3/2) - 3*sqrt(1/x^2 - 1))/((1/x^2 - 1)^3 + 3*(1/x^2 - 1)^2 + 3/x^2 - 2) + 3/8*((1/x^2 - 1)^(3/2) - sqrt(1/x^2 - 1))/((1/x^2 - 1)^2 + 2/ x^2 - 1) - 35/16*arctan(sqrt(1/x^2 - 1))
Time = 0.36 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, x^{2} \mathrm {sgn}\left (x\right ) - 19 \, \mathrm {sgn}\left (x\right )\right )} x^{2} + 87 \, \mathrm {sgn}\left (x\right )\right )} \sqrt {-x^{2} + 1} x + \frac {35}{16} \, \arcsin \left (x\right ) \mathrm {sgn}\left (x\right ) - \frac {x \mathrm {sgn}\left (x\right )}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\left (x\right )}{2 \, x} \]
1/48*(2*(4*x^2*sgn(x) - 19*sgn(x))*x^2 + 87*sgn(x))*sqrt(-x^2 + 1)*x + 35/ 16*arcsin(x)*sgn(x) - 1/2*x*sgn(x)/(sqrt(-x^2 + 1) - 1) + 1/2*(sqrt(-x^2 + 1) - 1)*sgn(x)/x
Time = 17.76 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx=\sqrt {\frac {1}{x^2}-1}-\frac {35\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{16}+\frac {19\,x^6\,\sqrt {\frac {1}{x^2}-1}}{16}+\frac {17\,x^6\,{\left (\frac {1}{x^2}-1\right )}^{3/2}}{6}+\frac {29\,x^6\,{\left (\frac {1}{x^2}-1\right )}^{5/2}}{16} \]