Integrand size = 18, antiderivative size = 37 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=-\frac {4}{3} \arctan \left (x^3+\sqrt {-1+x^6}\right )+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \] Output:
-4/3*arctan(x^3+(x^6-1)^(1/2))+1/3*ln(x^3+(x^6-1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=\frac {4}{3} \arctan \left (x^3-\sqrt {-1+x^6}\right )-\frac {1}{3} \log \left (-x^3+\sqrt {-1+x^6}\right ) \] Input:
Integrate[(-2 + x^3)/(x*Sqrt[-1 + x^6]),x]
Output:
(4*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - Log[-x^3 + Sqrt[-1 + x^6]]/3
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1803, 25, 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.
\(\displaystyle \int \frac {x^3-2}{x \sqrt {x^6-1}} \, dx\) |
\(\Big \downarrow \) 1803 |
\(\displaystyle \frac {1}{3} \int -\frac {2-x^3}{x^3 \sqrt {x^6-1}}dx^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {2-x^3}{x^3 \sqrt {x^6-1}}dx^3\) |
\(\Big \downarrow \) 538 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{\sqrt {x^6-1}}dx^3-2 \int \frac {1}{x^3 \sqrt {x^6-1}}dx^3\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}-2 \int \frac {1}{x^3 \sqrt {x^6-1}}dx^3\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-2 \int \frac {1}{x^3 \sqrt {x^6-1}}dx^3\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\int \frac {1}{x^3 \sqrt {x^6-1}}dx^6\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-2 \int \frac {1}{\sqrt {x^6-1}+1}d\sqrt {x^6-1}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-2 \arctan \left (\sqrt {x^6-1}\right )\right )\) |
Input:
Int[(-2 + x^3)/(x*Sqrt[-1 + x^6]),x]
Output:
(-2*ArcTan[Sqrt[-1 + x^6]] + ArcTanh[x^3/Sqrt[-1 + x^6]])/3
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_))/((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 = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {2 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(26\) |
trager | \(-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(47\) |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(86\) |
Input:
int((x^3-2)/x/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*ln(x^3+(x^6-1)^(1/2))+2/3*arctan(1/(x^6-1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=-\frac {4}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \] Input:
integrate((x^3-2)/x/(x^6-1)^(1/2),x, algorithm="fricas")
Output:
-4/3*arctan(-x^3 + sqrt(x^6 - 1)) - 1/3*log(-x^3 + sqrt(x^6 - 1))
Time = 1.97 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=- \frac {2 \left (\begin {cases} \operatorname {acos}{\left (\frac {1}{x^{3}} \right )} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}\right )}{3} + \frac {\log {\left (2 x^{3} + 2 \sqrt {x^{6} - 1} \right )}}{3} \] Input:
integrate((x**3-2)/x/(x**6-1)**(1/2),x)
Output:
-2*Piecewise((acos(x**(-3)), (x**3 > -1) & (x**3 < 1)))/3 + log(2*x**3 + 2 *sqrt(x**6 - 1))/3
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=-\frac {2}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \] Input:
integrate((x^3-2)/x/(x^6-1)^(1/2),x, algorithm="maxima")
Output:
-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)
\[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=\int { \frac {x^{3} - 2}{\sqrt {x^{6} - 1} x} \,d x } \] Input:
integrate((x^3-2)/x/(x^6-1)^(1/2),x, algorithm="giac")
Output:
integrate((x^3 - 2)/(sqrt(x^6 - 1)*x), x)
Time = 7.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=\frac {\ln \left (\sqrt {x^6-1}+x^3\right )}{3}-\frac {2\,\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \] Input:
int((x^3 - 2)/(x*(x^6 - 1)^(1/2)),x)
Output:
log((x^6 - 1)^(1/2) + x^3)/3 - (2*atan((x^6 - 1)^(1/2)))/3
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.65 \[ \int \frac {-2+x^3}{x \sqrt {-1+x^6}} \, dx=-\frac {\mathit {atan} \left (\frac {\sqrt {x^{6}-1}\, x^{6}-2 \sqrt {x^{6}-1}}{2 x^{6}-2}\right )}{3}+\frac {\mathrm {log}\left (\sqrt {x^{6}-1}+x^{3}\right )}{6}-\frac {\mathrm {log}\left (\sqrt {x^{6}-1}-x^{3}\right )}{6} \] Input:
int((x^3-2)/x/(x^6-1)^(1/2),x)
Output:
( - 2*atan((sqrt(x**6 - 1)*x**6 - 2*sqrt(x**6 - 1))/(2*x**6 - 2)) + log(sq rt(x**6 - 1) + x**3) - log(sqrt(x**6 - 1) - x**3))/6