Integrand size = 13, antiderivative size = 47 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{4 x^3}+\frac {1}{4} \arctan \left (\sqrt {-1+x^3}\right ) \] Output:
1/6*(x^3-1)^(1/2)/x^6+1/4*(x^3-1)^(1/2)/x^3+1/4*arctan((x^3-1)^(1/2))
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3} \left (2+3 x^3\right )}{12 x^6}+\frac {1}{4} \arctan \left (\sqrt {-1+x^3}\right ) \] Input:
Integrate[1/(x^7*Sqrt[-1 + x^3]),x]
Output:
(Sqrt[-1 + x^3]*(2 + 3*x^3))/(12*x^6) + ArcTan[Sqrt[-1 + x^3]]/4
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 52, 52, 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.
\(\displaystyle \int \frac {1}{x^7 \sqrt {x^3-1}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^9 \sqrt {x^3-1}}dx^3\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \int \frac {1}{x^6 \sqrt {x^3-1}}dx^3+\frac {\sqrt {x^3-1}}{2 x^6}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{x^3 \sqrt {x^3-1}}dx^3+\frac {\sqrt {x^3-1}}{x^3}\right )+\frac {\sqrt {x^3-1}}{2 x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \left (\int \frac {1}{x^6+1}d\sqrt {x^3-1}+\frac {\sqrt {x^3-1}}{x^3}\right )+\frac {\sqrt {x^3-1}}{2 x^6}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \left (\arctan \left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{x^3}\right )+\frac {\sqrt {x^3-1}}{2 x^6}\right )\) |
Input:
Int[1/(x^7*Sqrt[-1 + x^3]),x]
Output:
(Sqrt[-1 + x^3]/(2*x^6) + (3*(Sqrt[-1 + x^3]/x^3 + ArcTan[Sqrt[-1 + x^3]]) )/4)/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[(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]]
Time = 1.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {\sqrt {x^{3}-1}}{4 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{4}\) | \(36\) |
risch | \(\frac {3 x^{6}-x^{3}-2}{12 x^{6} \sqrt {x^{3}-1}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{4}\) | \(36\) |
elliptic | \(\frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {\sqrt {x^{3}-1}}{4 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{4}\) | \(36\) |
pseudoelliptic | \(\frac {3 \arctan \left (\sqrt {x^{3}-1}\right ) x^{6}+3 x^{3} \sqrt {x^{3}-1}+2 \sqrt {x^{3}-1}}{12 x^{6}}\) | \(41\) |
trager | \(\frac {\left (3 x^{3}+2\right ) \sqrt {x^{3}-1}}{12 x^{6}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{8}\) | \(63\) |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {\sqrt {\pi }}{2 x^{6}}-\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}+8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (12 x^{3}+8\right ) \sqrt {-x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{4}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\) | \(123\) |
Input:
int(1/x^7/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/6*(x^3-1)^(1/2)/x^6+1/4*(x^3-1)^(1/2)/x^3+1/4*arctan((x^3-1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {3 \, x^{6} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (3 \, x^{3} + 2\right )} \sqrt {x^{3} - 1}}{12 \, x^{6}} \] Input:
integrate(1/x^7/(x^3-1)^(1/2),x, algorithm="fricas")
Output:
1/12*(3*x^6*arctan(sqrt(x^3 - 1)) + (3*x^3 + 2)*sqrt(x^3 - 1))/x^6
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.94 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} - \frac {i}{4 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{12 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{6 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} + \frac {1}{4 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{12 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{2}} \sqrt {1 - \frac {1}{x^{3}}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**7/(x**3-1)**(1/2),x)
Output:
Piecewise((I*acosh(x**(-3/2))/4 - I/(4*x**(3/2)*sqrt(-1 + x**(-3))) + I/(1 2*x**(9/2)*sqrt(-1 + x**(-3))) + I/(6*x**(15/2)*sqrt(-1 + x**(-3))), 1/Abs (x**3) > 1), (-asin(x**(-3/2))/4 + 1/(4*x**(3/2)*sqrt(1 - 1/x**3)) - 1/(12 *x**(9/2)*sqrt(1 - 1/x**3)) - 1/(6*x**(15/2)*sqrt(1 - 1/x**3)), True))
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac {1}{4} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \] Input:
integrate(1/x^7/(x^3-1)^(1/2),x, algorithm="maxima")
Output:
1/12*(3*(x^3 - 1)^(3/2) + 5*sqrt(x^3 - 1))/(2*x^3 + (x^3 - 1)^2 - 1) + 1/4 *arctan(sqrt(x^3 - 1))
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, x^{6}} + \frac {1}{4} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \] Input:
integrate(1/x^7/(x^3-1)^(1/2),x, algorithm="giac")
Output:
1/12*(3*(x^3 - 1)^(3/2) + 5*sqrt(x^3 - 1))/x^6 + 1/4*arctan(sqrt(x^3 - 1))
Time = 0.15 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.02 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {x^3-1}}{4\,x^3}+\frac {\sqrt {x^3-1}}{6\,x^6}-\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(1/(x^7*(x^3 - 1)^(1/2)),x)
Output:
(x^3 - 1)^(1/2)/(4*x^3) + (x^3 - 1)^(1/2)/(6*x^6) - (3*((3^(1/2)*1i)/2 + 3 /2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1 /2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i)/2 + 3/2, asin((-(x - 1)/((3^(1/2)*1i) /2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(4*((( 3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*(( 3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^7 \sqrt {-1+x^3}} \, dx=\frac {3 \mathit {atan} \left (\frac {\sqrt {x^{3}-1}\, x^{3}-2 \sqrt {x^{3}-1}}{2 x^{3}-2}\right ) x^{6}+6 \sqrt {x^{3}-1}\, x^{3}+4 \sqrt {x^{3}-1}}{24 x^{6}} \] Input:
int(1/x^7/(x^3-1)^(1/2),x)
Output:
(3*atan((sqrt(x**3 - 1)*x**3 - 2*sqrt(x**3 - 1))/(2*x**3 - 2))*x**6 + 6*sq rt(x**3 - 1)*x**3 + 4*sqrt(x**3 - 1))/(24*x**6)