\(\int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx\) [463]

Optimal result

Integrand size = 15, antiderivative size = 71 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=-\frac {\sqrt {1-x^3}}{9 x^9}-\frac {5 \sqrt {1-x^3}}{36 x^6}-\frac {5 \sqrt {1-x^3}}{24 x^3}-\frac {5}{24} \text {arctanh}\left (\sqrt {1-x^3}\right ) \]

[Out]

-5/24*arctanh((-x^3+1)^(1/2))-1/9*(-x^3+1)^(1/2)/x^9-5/36*(-x^3+1)^(1/2)/x^6-5/24*(-x^3+1)^(1/2)/x^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65, 212} \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=-\frac {5}{24} \text {arctanh}\left (\sqrt {1-x^3}\right )-\frac {5 \sqrt {1-x^3}}{24 x^3}-\frac {\sqrt {1-x^3}}{9 x^9}-\frac {5 \sqrt {1-x^3}}{36 x^6} \]

[In]

Int[1/(x^10*Sqrt[1 - x^3]),x]

[Out]

-1/9*Sqrt[1 - x^3]/x^9 - (5*Sqrt[1 - x^3])/(36*x^6) - (5*Sqrt[1 - x^3])/(24*x^3) - (5*ArcTanh[Sqrt[1 - x^3]])/
24

Rule 44

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^4} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1-x^3}}{9 x^9}+\frac {5}{18} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^3} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1-x^3}}{9 x^9}-\frac {5 \sqrt {1-x^3}}{36 x^6}+\frac {5}{24} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1-x^3}}{9 x^9}-\frac {5 \sqrt {1-x^3}}{36 x^6}-\frac {5 \sqrt {1-x^3}}{24 x^3}+\frac {5}{48} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1-x^3}}{9 x^9}-\frac {5 \sqrt {1-x^3}}{36 x^6}-\frac {5 \sqrt {1-x^3}}{24 x^3}-\frac {5}{24} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^3}\right ) \\ & = -\frac {\sqrt {1-x^3}}{9 x^9}-\frac {5 \sqrt {1-x^3}}{36 x^6}-\frac {5 \sqrt {1-x^3}}{24 x^3}-\frac {5}{24} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=\frac {\sqrt {1-x^3} \left (-8-10 x^3-15 x^6\right )}{72 x^9}-\frac {5}{24} \text {arctanh}\left (\sqrt {1-x^3}\right ) \]

[In]

Integrate[1/(x^10*Sqrt[1 - x^3]),x]

[Out]

(Sqrt[1 - x^3]*(-8 - 10*x^3 - 15*x^6))/(72*x^9) - (5*ArcTanh[Sqrt[1 - x^3]])/24

Maple [A] (verified)

Time = 4.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.63

method	result	size
risch	\(\frac {15 x^{9}-5 x^{6}-2 x^{3}-8}{72 x^{9} \sqrt {-x^{3}+1}}-\frac {5 \,\operatorname {arctanh}\left (\sqrt {-x^{3}+1}\right )}{24}\)	\(45\)
trager	\(-\frac {\left (15 x^{6}+10 x^{3}+8\right ) \sqrt {-x^{3}+1}}{72 x^{9}}-\frac {5 \ln \left (-\frac {-x^{3}+2 \sqrt {-x^{3}+1}+2}{x^{3}}\right )}{48}\)	\(54\)
default	\(-\frac {5 \,\operatorname {arctanh}\left (\sqrt {-x^{3}+1}\right )}{24}-\frac {\sqrt {-x^{3}+1}}{9 x^{9}}-\frac {5 \sqrt {-x^{3}+1}}{36 x^{6}}-\frac {5 \sqrt {-x^{3}+1}}{24 x^{3}}\)	\(56\)
elliptic	\(-\frac {5 \,\operatorname {arctanh}\left (\sqrt {-x^{3}+1}\right )}{24}-\frac {\sqrt {-x^{3}+1}}{9 x^{9}}-\frac {5 \sqrt {-x^{3}+1}}{36 x^{6}}-\frac {5 \sqrt {-x^{3}+1}}{24 x^{3}}\)	\(56\)
pseudoelliptic	\(\frac {-15 \ln \left (-1+\sqrt {-x^{3}+1}\right ) x^{9}+15 \ln \left (1+\sqrt {-x^{3}+1}\right ) x^{9}+\left (30 x^{6}+20 x^{3}+16\right ) \sqrt {-x^{3}+1}}{144 \left (-1+\sqrt {-x^{3}+1}\right )^{3} \left (1+\sqrt {-x^{3}+1}\right )^{3}}\)	\(86\)
meijerg	\(-\frac {\frac {\sqrt {\pi }}{3 x^{9}}+\frac {\sqrt {\pi }}{4 x^{6}}+\frac {3 \sqrt {\pi }}{8 x^{3}}-\frac {5 \left (\frac {37}{30}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }\, \left (-148 x^{9}+144 x^{6}+96 x^{3}+128\right )}{384 x^{9}}+\frac {\sqrt {\pi }\, \left (240 x^{6}+160 x^{3}+128\right ) \sqrt {-x^{3}+1}}{384 x^{9}}+\frac {5 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{8}}{3 \sqrt {\pi }}\)	\(123\)

[In]

int(1/x^10/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/72*(15*x^9-5*x^6-2*x^3-8)/x^9/(-x^3+1)^(1/2)-5/24*arctanh((-x^3+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=-\frac {15 \, x^{9} \log \left (\sqrt {-x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt {-x^{3} + 1} - 1\right ) + 2 \, {\left (15 \, x^{6} + 10 \, x^{3} + 8\right )} \sqrt {-x^{3} + 1}}{144 \, x^{9}} \]

[In]

integrate(1/x^10/(-x^3+1)^(1/2),x, algorithm="fricas")

[Out]

-1/144*(15*x^9*log(sqrt(-x^3 + 1) + 1) - 15*x^9*log(sqrt(-x^3 + 1) - 1) + 2*(15*x^6 + 10*x^3 + 8)*sqrt(-x^3 +
1))/x^9

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.66 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.56 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=\begin {cases} - \frac {5 \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} + \frac {5}{24 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {5}{72 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {1}{36 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {1}{9 x^{\frac {21}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {5 i \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} - \frac {5 i}{24 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} + \frac {5 i}{72 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} + \frac {i}{36 x^{\frac {15}{2}} \sqrt {1 - \frac {1}{x^{3}}}} + \frac {i}{9 x^{\frac {21}{2}} \sqrt {1 - \frac {1}{x^{3}}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**10/(-x**3+1)**(1/2),x)

[Out]

Piecewise((-5*acosh(x**(-3/2))/24 + 5/(24*x**(3/2)*sqrt(-1 + x**(-3))) - 5/(72*x**(9/2)*sqrt(-1 + x**(-3))) -
1/(36*x**(15/2)*sqrt(-1 + x**(-3))) - 1/(9*x**(21/2)*sqrt(-1 + x**(-3))), 1/Abs(x**3) > 1), (5*I*asin(x**(-3/2
))/24 - 5*I/(24*x**(3/2)*sqrt(1 - 1/x**3)) + 5*I/(72*x**(9/2)*sqrt(1 - 1/x**3)) + I/(36*x**(15/2)*sqrt(1 - 1/x
**3)) + I/(9*x**(21/2)*sqrt(1 - 1/x**3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=-\frac {15 \, {\left (-x^{3} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-x^{3} + 1}}{72 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {5}{48} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {5}{48} \, \log \left (\sqrt {-x^{3} + 1} - 1\right ) \]

[In]

integrate(1/x^10/(-x^3+1)^(1/2),x, algorithm="maxima")

[Out]

-1/72*(15*(-x^3 + 1)^(5/2) - 40*(-x^3 + 1)^(3/2) + 33*sqrt(-x^3 + 1))/((x^3 - 1)^3 + 3*x^3 + 3*(x^3 - 1)^2 - 2
) - 5/48*log(sqrt(-x^3 + 1) + 1) + 5/48*log(sqrt(-x^3 + 1) - 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=-\frac {15 \, {\left (x^{3} - 1\right )}^{2} \sqrt {-x^{3} + 1} - 40 \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-x^{3} + 1}}{72 \, x^{9}} - \frac {5}{48} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {5}{48} \, \log \left ({\left | \sqrt {-x^{3} + 1} - 1 \right |}\right ) \]

[In]

integrate(1/x^10/(-x^3+1)^(1/2),x, algorithm="giac")

[Out]

-1/72*(15*(x^3 - 1)^2*sqrt(-x^3 + 1) - 40*(-x^3 + 1)^(3/2) + 33*sqrt(-x^3 + 1))/x^9 - 5/48*log(sqrt(-x^3 + 1)
+ 1) + 5/48*log(abs(sqrt(-x^3 + 1) - 1))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.14 \[ \int \frac {1}{x^{10} \sqrt {1-x^3}} \, dx=-\frac {5\,\sqrt {1-x^3}}{24\,x^3}-\frac {5\,\sqrt {1-x^3}}{36\,x^6}-\frac {\sqrt {1-x^3}}{9\,x^9}-\frac {5\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{8\,\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int(1/(x^10*(1 - x^3)^(1/2)),x)

[Out]

- (5*(1 - x^3)^(1/2))/(24*x^3) - (5*(1 - x^3)^(1/2))/(36*x^6) - (1 - x^3)^(1/2)/(9*x^9) - (5*((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 + (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)))/(8*(1 - x^3)^(1/2)*(((
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))