\(\int \frac {\sqrt {-1+x^3}}{x^7} \, dx\) [450]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {\left (-2+x^3\right ) \sqrt {-1+x^3}}{12 x^6}+\frac {1}{12} \arctan \left (\sqrt {-1+x^3}\right ) \]

[Out]

1/12*(x^3-2)*(x^3-1)^(1/2)/x^6+1/12*arctan((x^3-1)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 43, 44, 65, 209} \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {1}{12} \arctan \left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{12 x^3}-\frac {\sqrt {x^3-1}}{6 x^6} \]

[In]

Int[Sqrt[-1 + x^3]/x^7,x]

[Out]

-1/6*Sqrt[-1 + x^3]/x^6 + Sqrt[-1 + x^3]/(12*x^3) + ArcTan[Sqrt[-1 + x^3]]/12

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 {\sqrt {-1+x}}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{12 x^3}+\frac {1}{24} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{12 x^3}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right ) \\ & = -\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {\sqrt {-1+x^3}}{12 x^3}+\frac {1}{12} \arctan \left (\sqrt {-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {1}{12} \left (\frac {\left (-2+x^3\right ) \sqrt {-1+x^3}}{x^6}+\arctan \left (\sqrt {-1+x^3}\right )\right ) \]

[In]

Integrate[Sqrt[-1 + x^3]/x^7,x]

[Out]

(((-2 + x^3)*Sqrt[-1 + x^3])/x^6 + ArcTan[Sqrt[-1 + x^3]])/12

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {x^{6}-3 x^{3}+2}{12 x^{6} \sqrt {x^{3}-1}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(34\)
default	\(-\frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {\sqrt {x^{3}-1}}{12 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(36\)
elliptic	\(-\frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {\sqrt {x^{3}-1}}{12 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(36\)
pseudoelliptic	\(\frac {\arctan \left (\sqrt {x^{3}-1}\right ) x^{6}+x^{3} \sqrt {x^{3}-1}-2 \sqrt {x^{3}-1}}{12 x^{6}}\)	\(39\)
trager	\(\frac {\left (x^{3}-2\right ) \sqrt {x^{3}-1}}{12 x^{6}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{24}\)	\(61\)
meijerg	\(-\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{6}-8 x^{3}+8\right )}{8 x^{6}}+\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right ) \sqrt {-x^{3}+1}}{8 x^{6}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{6}}-\frac {\sqrt {\pi }}{x^{3}}\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\)	\(120\)

[In]

int((x^3-1)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/12*(x^6-3*x^3+2)/x^6/(x^3-1)^(1/2)+1/12*arctan((x^3-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {x^{6} \arctan \left (\sqrt {x^{3} - 1}\right ) + \sqrt {x^{3} - 1} {\left (x^{3} - 2\right )}}{12 \, x^{6}} \]

[In]

integrate((x^3-1)^(1/2)/x^7,x, algorithm="fricas")

[Out]

1/12*(x^6*arctan(sqrt(x^3 - 1)) + sqrt(x^3 - 1)*(x^3 - 2))/x^6

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.83 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{12} - \frac {i}{12 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{4 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {i}{6 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{12} + \frac {1}{12 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{4 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} + \frac {1}{6 x^{\frac {15}{2}} \sqrt {1 - \frac {1}{x^{3}}}} & \text {otherwise} \end {cases} \]

[In]

integrate((x**3-1)**(1/2)/x**7,x)

[Out]

Piecewise((I*acosh(x**(-3/2))/12 - I/(12*x**(3/2)*sqrt(-1 + x**(-3))) + I/(4*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))/12 + 1/(12*x**(3/2)*sqrt(1 - 1/x**3)) -
1/(4*x**(9/2)*sqrt(1 - 1/x**3)) + 1/(6*x**(15/2)*sqrt(1 - 1/x**3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac {1}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]

[In]

integrate((x^3-1)^(1/2)/x^7,x, algorithm="maxima")

[Out]

1/12*((x^3 - 1)^(3/2) - sqrt(x^3 - 1))/(2*x^3 + (x^3 - 1)^2 - 1) + 1/12*arctan(sqrt(x^3 - 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (28) = 56\).

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {1}{48} \, \pi + \frac {\sqrt {x^{3} - 1} - \frac {1}{\sqrt {x^{3} - 1}}}{12 \, {\left ({\left (\sqrt {x^{3} - 1} - \frac {1}{\sqrt {x^{3} - 1}}\right )}^{2} + 4\right )}} + \frac {1}{24} \, \arctan \left (\frac {x^{3} - 2}{2 \, \sqrt {x^{3} - 1}}\right ) \]

[In]

integrate((x^3-1)^(1/2)/x^7,x, algorithm="giac")

[Out]

1/48*pi + 1/12*(sqrt(x^3 - 1) - 1/sqrt(x^3 - 1))/((sqrt(x^3 - 1) - 1/sqrt(x^3 - 1))^2 + 4) + 1/24*arctan(1/2*(
x^3 - 2)/sqrt(x^3 - 1))

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 189, normalized size of antiderivative = 5.25 \[ \int \frac {\sqrt {-1+x^3}}{x^7} \, dx=\frac {\sqrt {x^3-1}}{12\,x^3}-\frac {\sqrt {x^3-1}}{6\,x^6}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int((x^3 - 1)^(1/2)/x^7,x)

[Out]

(x^3 - 1)^(1/2)/(12*x^3) - (x^3 - 1)^(1/2)/(6*x^6) - (((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))