Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {\sqrt {-1-x^3}}{6 x^6}-\frac {\sqrt {-1-x^3}}{4 x^3}+\frac {1}{4} \arctan \left (\sqrt {-1-x^3}\right ) \] 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 = 42, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {\left (2-3 x^3\right ) \sqrt {-1-x^3}}{12 x^6}+\frac {1}{4} \arctan \left (\sqrt {-1-x^3}\right ) \] Input:
Integrate[1/(x^7*Sqrt[-1 - x^3]),x]
Output:
((2 - 3*x^3)*Sqrt[-1 - x^3])/(12*x^6) + ArcTan[Sqrt[-1 - x^3]]/4
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 52, 52, 73, 217}
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 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {-x^3-1}}{2 x^6}-\frac {3}{4} \int \frac {1}{x^6 \sqrt {-x^3-1}}dx^3\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {-x^3-1}}{2 x^6}-\frac {3}{4} \left (\frac {\sqrt {-x^3-1}}{x^3}-\frac {1}{2} \int \frac {1}{x^3 \sqrt {-x^3-1}}dx^3\right )\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {-x^3-1}}{2 x^6}-\frac {3}{4} \left (\int \frac {1}{-x^6-1}d\sqrt {-x^3-1}+\frac {\sqrt {-x^3-1}}{x^3}\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {-x^3-1}}{2 x^6}-\frac {3}{4} \left (\frac {\sqrt {-x^3-1}}{x^3}-\arctan \left (\sqrt {-x^3-1}\right )\right )\right )\) |
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[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 = 0.81 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {3 x^{6}+x^{3}-2}{12 x^{6} \sqrt {-x^{3}-1}}+\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{4}\) | \(38\) |
default | \(\frac {\sqrt {-x^{3}-1}}{6 x^{6}}-\frac {\sqrt {-x^{3}-1}}{4 x^{3}}+\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{4}\) | \(42\) |
elliptic | \(\frac {\sqrt {-x^{3}-1}}{6 x^{6}}-\frac {\sqrt {-x^{3}-1}}{4 x^{3}}+\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{4}\) | \(42\) |
pseudoelliptic | \(\frac {3 \arctan \left (\sqrt {-x^{3}-1}\right ) x^{6}-3 x^{3} \sqrt {-x^{3}-1}+2 \sqrt {-x^{3}-1}}{12 x^{6}}\) | \(47\) |
trager | \(-\frac {\left (3 x^{3}-2\right ) \sqrt {-x^{3}-1}}{12 x^{6}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {-x^{3}-1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{8}\) | \(67\) |
meijerg | \(-\frac {i \left (-\frac {\sqrt {\pi }}{2 x^{6}}+\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}-8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (-12 x^{3}+8\right ) \sqrt {x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )}{4}\right )}{3 \sqrt {\pi }}\) | \(98\) |
Input:
int(1/x^7/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/12*(3*x^6+x^3-2)/x^6/(-x^3-1)^(1/2)+1/4*arctan((-x^3-1)^(1/2))
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {3 \, x^{6} \arctan \left (\sqrt {-x^{3} - 1}\right ) - {\left (3 \, x^{3} - 2\right )} \sqrt {-x^{3} - 1}}{12 \, x^{6}} \] 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 = 2.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {i \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} - \frac {i}{4 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {i}{12 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {i}{6 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \] Input:
integrate(1/x**7/(-x**3-1)**(1/2),x)
Output:
I*asinh(x**(-3/2))/4 - I/(4*x**(3/2)*sqrt(1 + x**(-3))) - I/(12*x**(9/2)*s qrt(1 + x**(-3))) + I/(6*x**(15/2)*sqrt(1 + x**(-3)))
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=-\frac {3 \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {-x^{3} - 1}}{12 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{4} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \] 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 = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {3 \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {-x^{3} - 1}}{12 \, x^{6}} + \frac {1}{4} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \] 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 = 0.04 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.94 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {\sqrt {-x^3-1}}{6\,x^6}-\frac {\sqrt {-x^3-1}}{4\,x^3}-\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(1/(x^7*(- x^3 - 1)^(1/2)),x)
Output:
(- x^3 - 1)^(1/2)/(6*x^6) - (- x^3 - 1)^(1/2)/(4*x^3) - (3*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(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)*(((3^(1/2)*1i)/2 - x + 1/2) /((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*(- x^3 - 1)^(1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2) *1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {i \left (-6 \sqrt {x^{3}+1}\, x^{3}+4 \sqrt {x^{3}+1}-3 \,\mathrm {log}\left (\sqrt {x^{3}+1}-1\right ) x^{6}+3 \,\mathrm {log}\left (\sqrt {x^{3}+1}+1\right ) x^{6}\right )}{24 x^{6}} \] Input:
int(1/x^7/(-x^3-1)^(1/2),x)
Output:
(i*( - 6*sqrt(x**3 + 1)*x**3 + 4*sqrt(x**3 + 1) - 3*log(sqrt(x**3 + 1) - 1 )*x**6 + 3*log(sqrt(x**3 + 1) + 1)*x**6))/(24*x**6)