Integrand size = 18, antiderivative size = 59 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{6} \left (-2+x^3\right ) \sqrt {-1+x^6}-\frac {2}{3} \arctan \left (\frac {1+x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]
1/6*(x^3-2)*(x^6-1)^(1/2)-2/3*arctan((x^3+1)/(x^6-1)^(1/2))-1/3*arctanh((x ^3+1)/(x^6-1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{6} \left (\left (-2+x^3\right ) \sqrt {-1+x^6}-4 \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )-2 \text {arctanh}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )\right ) \]
((-2 + x^3)*Sqrt[-1 + x^6] - 4*ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)] - 2*ArcTa nh[Sqrt[-1 + x^6]/(-1 + x^3)])/6
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.81, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1396, 948, 112, 171, 25, 175, 43, 103, 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.
\(\displaystyle \int \frac {\left (x^3-1\right ) \sqrt {x^6-1}}{x} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {x^6-1} \int \frac {\left (x^3-1\right )^{3/2} \sqrt {x^3+1}}{x}dx}{\sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\sqrt {x^6-1} \int \frac {\left (x^3-1\right )^{3/2} \sqrt {x^3+1}}{x^3}dx^3}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 112 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (x^3-1\right )^{3/2} \sqrt {x^3+1}-\frac {1}{2} \int \frac {\sqrt {x^3-1} \left (x^3+2\right )}{x^3 \sqrt {x^3+1}}dx^3\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (-\int -\frac {2-x^3}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3-\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {1}{2} \sqrt {x^3+1} \left (x^3-1\right )^{3/2}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (\int \frac {2-x^3}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3-\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {1}{2} \sqrt {x^3+1} \left (x^3-1\right )^{3/2}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (-\int \frac {1}{\sqrt {x^3-1} \sqrt {x^3+1}}dx^3+2 \int \frac {1}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3-\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {1}{2} \sqrt {x^3+1} \left (x^3-1\right )^{3/2}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (2 \int \frac {1}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3-\text {arccosh}\left (x^3\right )-\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {1}{2} \sqrt {x^3+1} \left (x^3-1\right )^{3/2}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (2 \int \frac {1}{x^6+1}d\left (\sqrt {x^3-1} \sqrt {x^3+1}\right )-\text {arccosh}\left (x^3\right )-\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {1}{2} \sqrt {x^3+1} \left (x^3-1\right )^{3/2}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{2} \left (-\text {arccosh}\left (x^3\right )+2 \arctan \left (\sqrt {x^3-1} \sqrt {x^3+1}\right )-\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {1}{2} \sqrt {x^3+1} \left (x^3-1\right )^{3/2}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
(Sqrt[-1 + x^6]*(((-1 + x^3)^(3/2)*Sqrt[1 + x^3])/2 + (-(Sqrt[-1 + x^3]*Sq rt[1 + x^3]) - ArcCosh[x^3] + 2*ArcTan[Sqrt[-1 + x^3]*Sqrt[1 + x^3]])/2))/ (3*Sqrt[-1 + x^3]*Sqrt[1 + x^3])
3.8.64.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
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[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 0.98 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}-\frac {\sqrt {x^{6}-1}}{3}-\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(47\) |
trager | \(\left (\frac {x^{3}}{6}-\frac {1}{3}\right ) \sqrt {x^{6}-1}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}+\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{6}\) | \(62\) |
meijerg | \(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}+\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(136\) |
1/6*x^3*(x^6-1)^(1/2)-1/6*ln(x^3+(x^6-1)^(1/2))-1/3*(x^6-1)^(1/2)-1/3*arct an(1/(x^6-1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {1}{6} \, \sqrt {x^{6} - 1} {\left (x^{3} - 2\right )} + \frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]
1/6*sqrt(x^6 - 1)*(x^3 - 2) + 2/3*arctan(-x^3 + sqrt(x^6 - 1)) + 1/6*log(- x^3 + sqrt(x^6 - 1))
Time = 4.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {x^{3} \sqrt {x^{6} - 1}}{6} - \frac {\begin {cases} \sqrt {x^{6} - 1} - \operatorname {acos}{\left (\frac {1}{x^{3}} \right )} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} - \frac {\log {\left (2 x^{3} + 2 \sqrt {x^{6} - 1} \right )}}{6} \]
x**3*sqrt(x**6 - 1)/6 - Piecewise((sqrt(x**6 - 1) - acos(x**(-3)), (x**3 > -1) & (x**3 < 1)))/3 - log(2*x**3 + 2*sqrt(x**6 - 1))/6
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=-\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {\sqrt {x^{6} - 1}}{6 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} + \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) - \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
-1/3*sqrt(x^6 - 1) - 1/6*sqrt(x^6 - 1)/(x^3*((x^6 - 1)/x^6 - 1)) + 1/3*arc tan(sqrt(x^6 - 1)) - 1/12*log(sqrt(x^6 - 1)/x^3 + 1) + 1/12*log(sqrt(x^6 - 1)/x^3 - 1)
\[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{x} \,d x } \]
Time = 6.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x} \, dx=\frac {x^3\,\sqrt {x^6-1}}{6}-\frac {\sqrt {x^6-1}}{3}-\frac {\ln \left (\sqrt {x^6-1}+x^3\right )}{6}+\frac {\ln \left (\frac {\sqrt {x^6-1}+1{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{3} \]