3.9.48 \(\int \frac {(-1+x^3) \sqrt {-1+x^6}}{x^7} \, dx\) [848]

Optimal. Leaf size=64 \[ \frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}+\frac {1}{3} \text {ArcTan}\left (\frac {1+x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1489, 825, 858, 223, 212, 272, 65, 209} \begin {gather*} -\frac {1}{6} \text {ArcTan}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{6 x^6}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)*Sqrt[-1 + x^6])/x^7,x]

[Out]

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

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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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]]

Rule 825

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^
(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1489

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(-1+x) \sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right )\\ &=\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}+\frac {1}{12} \text {Subst}\left (\int \frac {-2+4 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 65, normalized size = 1.02 \begin {gather*} \frac {1}{6} \left (\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{x^6}+2 \text {ArcTan}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )+4 \tanh ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)*Sqrt[-1 + x^6])/x^7,x]

[Out]

(((1 - 2*x^3)*Sqrt[-1 + x^6])/x^6 + 2*ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)] + 4*ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)]
)/6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.99, size = 64, normalized size = 1.00

method	result	size
trager	\(-\frac {\left (2 x^{3}-1\right ) \sqrt {x^{6}-1}}{6 x^{6}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}\)	\(64\)
risch	\(-\frac {2 x^{9}-x^{6}-2 x^{3}+1}{6 x^{6} \sqrt {x^{6}-1}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\)	\(115\)
meijerg	\(-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{4 x^{6}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-\left (-2 \ln \left (2\right )-1+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-\frac {2 \sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}-\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{3}}-4 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)*(x^6-1)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*(2*x^3-1)/x^6*(x^6-1)^(1/2)-1/3*ln(-x^3+(x^6-1)^(1/2))-1/6*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)+(x^6-1)^(1/2
))/x^3)

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Maxima [A]
time = 0.46, size = 67, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 64, normalized size = 1.00 \begin {gather*} -\frac {2 \, x^{6} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{6} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{6} + \sqrt {x^{6} - 1} {\left (2 \, x^{3} - 1\right )}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 3.46, size = 151, normalized size = 2.36 \begin {gather*} - \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{6} + \frac {i}{6 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{6 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{6 x^{3}} & \text {otherwise} \end {cases} + \begin {cases} - \frac {x^{3}}{3 \sqrt {x^{6} - 1}} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} + \frac {1}{3 x^{3} \sqrt {x^{6} - 1}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {i x^{3}}{3 \sqrt {1 - x^{6}}} - \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} - \frac {i}{3 x^{3} \sqrt {1 - x^{6}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)*(x**6-1)**(1/2)/x**7,x)

[Out]

-Piecewise((I*acosh(x**(-3))/6 + I/(6*x**3*sqrt(-1 + x**(-6))) - I/(6*x**9*sqrt(-1 + x**(-6))), 1/Abs(x**6) >
1), (-asin(x**(-3))/6 - sqrt(1 - 1/x**6)/(6*x**3), True)) + Piecewise((-x**3/(3*sqrt(x**6 - 1)) + acosh(x**3)/
3 + 1/(3*x**3*sqrt(x**6 - 1)), Abs(x**6) > 1), (I*x**3/(3*sqrt(1 - x**6)) - I*asin(x**3)/3 - I/(3*x**3*sqrt(1
- x**6)), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^7,x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - 1)*(x^3 - 1)/x^7, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^3-1\right )\,\sqrt {x^6-1}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/x^7,x)

[Out]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/x^7, x)

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