Integrand size = 20, antiderivative size = 60 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=\frac {\left (-1+2 x^3\right ) \sqrt {-1+x^6}}{3 x^3}-\frac {4}{3} \arctan \left (x^3+\sqrt {-1+x^6}\right )+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=\frac {1}{3} \left (\frac {\left (-1+2 x^3\right ) \sqrt {-1+x^6}}{x^3}+4 \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )+2 \text {arctanh}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )\right ) \]
(((-1 + 2*x^3)*Sqrt[-1 + x^6])/x^3 + 4*ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)] + 2*ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)])/3
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1803, 536, 538, 224, 219, 243, 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.
\(\displaystyle \int \frac {\left (2 x^3+1\right ) \sqrt {x^6-1}}{x^4} \, dx\) |
\(\Big \downarrow \) 1803 |
\(\displaystyle \frac {1}{3} \int \frac {\left (2 x^3+1\right ) \sqrt {x^6-1}}{x^6}dx^3\) |
\(\Big \downarrow \) 536 |
\(\displaystyle \frac {1}{3} \left (\int \frac {x^3-2}{x^3 \sqrt {x^6-1}}dx^3-\frac {\left (1-2 x^3\right ) \sqrt {x^6-1}}{x^3}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{\sqrt {x^6-1}}dx^3-2 \int \frac {1}{x^3 \sqrt {x^6-1}}dx^3-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{x^3}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}-2 \int \frac {1}{x^3 \sqrt {x^6-1}}dx^3-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{x^3}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (-2 \int \frac {1}{x^3 \sqrt {x^6-1}}dx^3+\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{x^3}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{3} \left (-\int \frac {1}{x^3 \sqrt {x^6-1}}dx^6+\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{x^3}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-2 \int \frac {1}{\sqrt {x^6-1}+1}d\sqrt {x^6-1}+\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{x^3}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} \left (-2 \arctan \left (\sqrt {x^6-1}\right )+\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{x^3}\right )\) |
(-(((1 - 2*x^3)*Sqrt[-1 + x^6])/x^3) - 2*ArcTan[Sqrt[-1 + x^6]] + ArcTanh[ x^3/Sqrt[-1 + x^6]])/3
3.8.85.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x )^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Time = 1.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right ) x^{3}+2 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{3}+2 x^{3} \sqrt {x^{6}-1}-\sqrt {x^{6}-1}}{3 x^{3}}\) | \(57\) |
trager | \(\frac {\left (2 x^{3}-1\right ) \sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(64\) |
meijerg | \(-\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{3}}-4 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(136\) |
risch | \(-\frac {\sqrt {x^{6}-1}}{3 x^{3}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{6}+1}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(141\) |
1/3*(ln(x^3+(x^6-1)^(1/2))*x^3+2*arctan(1/(x^6-1)^(1/2))*x^3+2*x^3*(x^6-1) ^(1/2)-(x^6-1)^(1/2))/x^3
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=-\frac {4 \, x^{3} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} - \sqrt {x^{6} - 1} {\left (2 \, x^{3} - 1\right )}}{3 \, x^{3}} \]
-1/3*(4*x^3*arctan(-x^3 + sqrt(x^6 - 1)) + x^3*log(-x^3 + sqrt(x^6 - 1)) + x^3 - sqrt(x^6 - 1)*(2*x^3 - 1))/x^3
Result contains complex when optimal does not.
Time = 3.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.80 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=\begin {cases} - \frac {x^{3}}{3 \sqrt {x^{6} - 1}} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} + \frac {1}{3 x^{3} \sqrt {x^{6} - 1}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {i x^{3}}{3 \sqrt {1 - x^{6}}} - \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} - \frac {i}{3 x^{3} \sqrt {1 - x^{6}}} & \text {otherwise} \end {cases} + 2 \left (\begin {cases} - \frac {i x^{3}}{3 \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {i}{3 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {x^{3}}{3 \sqrt {1 - \frac {1}{x^{6}}}} + \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases}\right ) \]
Piecewise((-x**3/(3*sqrt(x**6 - 1)) + acosh(x**3)/3 + 1/(3*x**3*sqrt(x**6 - 1)), Abs(x**6) > 1), (I*x**3/(3*sqrt(1 - x**6)) - I*asin(x**3)/3 - I/(3* x**3*sqrt(1 - x**6)), True)) + 2*Piecewise((-I*x**3/(3*sqrt(-1 + x**(-6))) - I*acosh(x**(-3))/3 + I/(3*x**3*sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1), ( x**3/(3*sqrt(1 - 1/x**6)) + asin(x**(-3))/3 - 1/(3*x**3*sqrt(1 - 1/x**6)), True))
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=\frac {2}{3} \, \sqrt {x^{6} - 1} - \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {2}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
2/3*sqrt(x^6 - 1) - 1/3*sqrt(x^6 - 1)/x^3 - 2/3*arctan(sqrt(x^6 - 1)) + 1/ 6*log(sqrt(x^6 - 1)/x^3 + 1) - 1/6*log(sqrt(x^6 - 1)/x^3 - 1)
\[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (2 \, x^{3} + 1\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx=\int \frac {\sqrt {x^6-1}\,\left (2\,x^3+1\right )}{x^4} \,d x \]