Integrand size = 23, antiderivative size = 41 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=-\frac {2 \sqrt {-1+x^3} \left (-1+x^3+3 x^6\right )}{9 x^9}+\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \] Output:
-2/9*(x^3-1)^(1/2)*(3*x^6+x^3-1)/x^9+2/3*arctan((x^3-1)^(1/2))
Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=-\frac {2 \sqrt {-1+x^3} \left (-1+x^3+3 x^6\right )}{9 x^9}+\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \] Input:
Integrate[(Sqrt[-1 + x^3]*(-2 + x^3 + 2*x^6))/x^10,x]
Output:
(-2*Sqrt[-1 + x^3]*(-1 + x^3 + 3*x^6))/(9*x^9) + (2*ArcTan[Sqrt[-1 + x^3]] )/3
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(41)=82\).
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1802, 25, 1192, 25, 1580, 27, 1471, 27, 252, 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.
\(\displaystyle \int \frac {\sqrt {x^3-1} \left (2 x^6+x^3-2\right )}{x^{10}} \, dx\) |
\(\Big \downarrow \) 1802 |
\(\displaystyle \frac {1}{3} \int -\frac {\sqrt {x^3-1} \left (-2 x^6-x^3+2\right )}{x^{12}}dx^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {\sqrt {x^3-1} \left (-2 x^6-x^3+2\right )}{x^{12}}dx^3\) |
\(\Big \downarrow \) 1192 |
\(\displaystyle -\frac {2}{3} \int -\frac {x^6 \left (2 x^{12}+5 x^6+1\right )}{\left (x^6+1\right )^4}d\sqrt {x^3-1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} \int \frac {x^6 \left (2 x^{12}+5 x^6+1\right )}{\left (x^6+1\right )^4}d\sqrt {x^3-1}\) |
\(\Big \downarrow \) 1580 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{6} \int \frac {2 \left (-6 x^{12}-9 x^6+1\right )}{\left (x^6+1\right )^3}d\sqrt {x^3-1}-\frac {\sqrt {x^3-1}}{3 \left (x^6+1\right )^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{3} \int \frac {-6 x^{12}-9 x^6+1}{\left (x^6+1\right )^3}d\sqrt {x^3-1}-\frac {\sqrt {x^3-1}}{3 \left (x^6+1\right )^3}\right )\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{3} \left (\frac {\sqrt {x^3-1}}{\left (x^6+1\right )^2}-\frac {1}{4} \int \frac {24 x^6}{\left (x^6+1\right )^2}d\sqrt {x^3-1}\right )-\frac {\sqrt {x^3-1}}{3 \left (x^6+1\right )^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{3} \left (\frac {\sqrt {x^3-1}}{\left (x^6+1\right )^2}-6 \int \frac {x^6}{\left (x^6+1\right )^2}d\sqrt {x^3-1}\right )-\frac {\sqrt {x^3-1}}{3 \left (x^6+1\right )^3}\right )\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{3} \left (\frac {\sqrt {x^3-1}}{\left (x^6+1\right )^2}-6 \left (\frac {1}{2} \int \frac {1}{x^6+1}d\sqrt {x^3-1}-\frac {\sqrt {x^3-1}}{2 \left (x^6+1\right )}\right )\right )-\frac {\sqrt {x^3-1}}{3 \left (x^6+1\right )^3}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{3} \left (\frac {\sqrt {x^3-1}}{\left (x^6+1\right )^2}-6 \left (\frac {1}{2} \arctan \left (\sqrt {x^3-1}\right )-\frac {\sqrt {x^3-1}}{2 \left (x^6+1\right )}\right )\right )-\frac {\sqrt {x^3-1}}{3 \left (x^6+1\right )^3}\right )\) |
Input:
Int[(Sqrt[-1 + x^3]*(-2 + x^3 + 2*x^6))/x^10,x]
Output:
(-2*(-1/3*Sqrt[-1 + x^3]/(1 + x^6)^3 + (Sqrt[-1 + x^3]/(1 + x^6)^2 - 6*(-1 /2*Sqrt[-1 + x^3]/(1 + x^6) + ArcTan[Sqrt[-1 + x^3]]/2))/3))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* (q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b *d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e }, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1 )/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.40 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(\frac {6 \arctan \left (\sqrt {x^{3}-1}\right ) x^{9}+\left (-6 x^{6}-2 x^{3}+2\right ) \sqrt {x^{3}-1}}{9 x^{9}}\) | \(40\) |
risch | \(-\frac {2 \left (3 x^{9}-2 x^{6}-2 x^{3}+1\right )}{9 x^{9} \sqrt {x^{3}-1}}+\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(41\) |
default | \(-\frac {2 \sqrt {x^{3}-1}}{9 x^{6}}+\frac {2 \sqrt {x^{3}-1}}{9 x^{9}}-\frac {2 \sqrt {x^{3}-1}}{3 x^{3}}+\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(48\) |
elliptic | \(-\frac {2 \sqrt {x^{3}-1}}{9 x^{6}}+\frac {2 \sqrt {x^{3}-1}}{9 x^{9}}-\frac {2 \sqrt {x^{3}-1}}{3 x^{3}}+\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(48\) |
trager | \(-\frac {2 \sqrt {x^{3}-1}\, \left (3 x^{6}+x^{3}-1\right )}{9 x^{9}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}-1}}{x^{3}}\right )}{3}\) | \(66\) |
meijerg | \(\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {2 \sqrt {\pi }}{x^{3}}-\left (-2 \ln \left (2\right )-1+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{4 x^{3}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )\right )}{3 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}-\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }}{x^{6}}-\frac {\sqrt {\pi }}{x^{3}}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (x^{6}-8 x^{3}+8\right )}{8 x^{6}}+\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right ) \sqrt {-x^{3}+1}}{8 x^{6}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{2}\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}-\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {2 \sqrt {\pi }}{3 x^{9}}+\frac {\sqrt {\pi }}{2 x^{6}}+\frac {\sqrt {\pi }}{4 x^{3}}-\frac {\left (\frac {5}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (20 x^{9}-48 x^{6}-96 x^{3}+128\right )}{192 x^{9}}-\frac {\sqrt {\pi }\, \left (-48 x^{6}-32 x^{3}+128\right ) \sqrt {-x^{3}+1}}{192 x^{9}}+\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{4}\right )}{3 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\) | \(363\) |
Input:
int((x^3-1)^(1/2)*(2*x^6+x^3-2)/x^10,x,method=_RETURNVERBOSE)
Output:
1/9*(6*arctan((x^3-1)^(1/2))*x^9+(-6*x^6-2*x^3+2)*(x^3-1)^(1/2))/x^9
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=\frac {2 \, {\left (3 \, x^{9} \arctan \left (\sqrt {x^{3} - 1}\right ) - {\left (3 \, x^{6} + x^{3} - 1\right )} \sqrt {x^{3} - 1}\right )}}{9 \, x^{9}} \] Input:
integrate((x^3-1)^(1/2)*(2*x^6+x^3-2)/x^10,x, algorithm="fricas")
Output:
2/9*(3*x^9*arctan(sqrt(x^3 - 1)) - (3*x^6 + x^3 - 1)*sqrt(x^3 - 1))/x^9
Time = 50.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=\frac {2 \operatorname {atan}{\left (\sqrt {x^{3} - 1} \right )}}{3} - \frac {2 \sqrt {x^{3} - 1}}{3 x^{3}} - \frac {2 \left (x^{3} - 1\right )^{\frac {3}{2}}}{9 x^{9}} \] Input:
integrate((x**3-1)**(1/2)*(2*x**6+x**3-2)/x**10,x)
Output:
2*atan(sqrt(x**3 - 1))/3 - 2*sqrt(x**3 - 1)/(3*x**3) - 2*(x**3 - 1)**(3/2) /(9*x**9)
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (33) = 66\).
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.76 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=-\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{3} - 1}}{36 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} + \frac {{\left (x^{3} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {2 \, \sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \] Input:
integrate((x^3-1)^(1/2)*(2*x^6+x^3-2)/x^10,x, algorithm="maxima")
Output:
-1/36*(3*(x^3 - 1)^(5/2) + 8*(x^3 - 1)^(3/2) - 3*sqrt(x^3 - 1))/((x^3 - 1) ^3 + 3*x^3 + 3*(x^3 - 1)^2 - 2) + 1/12*((x^3 - 1)^(3/2) - sqrt(x^3 - 1))/( 2*x^3 + (x^3 - 1)^2 - 1) - 2/3*sqrt(x^3 - 1)/x^3 + 2/3*arctan(sqrt(x^3 - 1 ))
Time = 0.48 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=-\frac {2 \, {\left (3 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 7 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {x^{3} - 1}\right )}}{9 \, x^{9}} + \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \] Input:
integrate((x^3-1)^(1/2)*(2*x^6+x^3-2)/x^10,x, algorithm="giac")
Output:
-2/9*(3*(x^3 - 1)^(5/2) + 7*(x^3 - 1)^(3/2) + 3*sqrt(x^3 - 1))/x^9 + 2/3*a rctan(sqrt(x^3 - 1))
Time = 7.45 (sec) , antiderivative size = 201, normalized size of antiderivative = 4.90 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=\frac {2\,\sqrt {x^3-1}}{9\,x^9}-\frac {2\,\sqrt {x^3-1}}{9\,x^6}-\frac {2\,\sqrt {x^3-1}}{3\,x^3}-\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(((x^3 - 1)^(1/2)*(x^3 + 2*x^6 - 2))/x^10,x)
Output:
(2*(x^3 - 1)^(1/2))/(9*x^9) - (2*(x^3 - 1)^(1/2))/(9*x^6) - (2*(x^3 - 1)^( 1/2))/(3*x^3) - (2*((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)))/(((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 = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {-1+x^3} \left (-2+x^3+2 x^6\right )}{x^{10}} \, dx=\frac {3 \mathit {atan} \left (\frac {\sqrt {x^{3}-1}\, x^{3}-2 \sqrt {x^{3}-1}}{2 x^{3}-2}\right ) x^{9}-6 \sqrt {x^{3}-1}\, x^{6}-2 \sqrt {x^{3}-1}\, x^{3}+2 \sqrt {x^{3}-1}}{9 x^{9}} \] Input:
int((x^3-1)^(1/2)*(2*x^6+x^3-2)/x^10,x)
Output:
(3*atan((sqrt(x**3 - 1)*x**3 - 2*sqrt(x**3 - 1))/(2*x**3 - 2))*x**9 - 6*sq rt(x**3 - 1)*x**6 - 2*sqrt(x**3 - 1)*x**3 + 2*sqrt(x**3 - 1))/(9*x**9)