3.8.85 \(\int \frac {(1+2 x^3) \sqrt {-1+x^6}}{x^4} \, dx\)

Optimal. Leaf size=60 \[ \frac {\sqrt {x^6-1} \left (2 x^3-1\right )}{3 x^3}+\frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )-\frac {4}{3} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 56, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1475, 813, 844, 217, 206, 266, 63, 203} \begin {gather*} -\frac {2}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{3 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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 266

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 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

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 1475

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 63, normalized size = 1.05 \begin {gather*} \frac {1}{3} \left (-2 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (2 x^3-1\right )}{x^3}-\frac {\sqrt {x^6-1} \sin ^{-1}\left (x^3\right )}{\sqrt {1-x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.18, size = 68, normalized size = 1.13 \begin {gather*} \frac {\left (-1+2 x^3\right ) \sqrt {-1+x^6}}{3 x^3}+\frac {4}{3} \tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 62, normalized size = 1.03 \begin {gather*} -\frac {4 \, x^{3} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} - \sqrt {x^{6} - 1} {\left (2 \, x^{3} - 1\right )}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+1)*(x^6-1)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/3*(4*x^3*arctan(-x^3 + sqrt(x^6 - 1)) + x^3*log(-x^3 + sqrt(x^6 - 1)) + x^3 - sqrt(x^6 - 1)*(2*x^3 - 1))/x^
3

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - 1} {\left (2 \, x^{3} + 1\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+1)*(x^6-1)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - 1)*(2*x^3 + 1)/x^4, x)

________________________________________________________________________________________

maple [C]  time = 0.53, size = 64, normalized size = 1.07

method	result	size
trager	\(\frac {\left (2 x^{3}-1\right ) \sqrt {x^{6}-1}}{3 x^{3}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\)	\(64\)
meijerg	\(-\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{3}}-4 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}\)	\(136\)
risch	\(-\frac {\sqrt {x^{6}-1}}{3 x^{3}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{6}+1}\right )}{3 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }\right )}{3 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+1)*(x^6-1)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 64, normalized size = 1.07 \begin {gather*} \frac {2}{3} \, \sqrt {x^{6} - 1} - \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {2}{3} \, \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((2*x^3+1)*(x^6-1)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x^6-1}\,\left (2\,x^3+1\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)^(1/2)*(2*x^3 + 1))/x^4,x)

[Out]

int(((x^6 - 1)^(1/2)*(2*x^3 + 1))/x^4, x)

________________________________________________________________________________________

sympy [C]  time = 5.22, size = 168, normalized size = 2.80 \begin {gather*} \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} + 2 \left (\begin {cases} - \frac {i x^{3}}{3 \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {i}{3 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {x^{3}}{3 \sqrt {1 - \frac {1}{x^{6}}}} + \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+1)*(x**6-1)**(1/2)/x**4,x)

[Out]

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*sq
rt(1 - x**6)) - I*asin(x**3)/3 - I/(3*x**3*sqrt(1 - x**6)), True)) + 2*Piecewise((-I*x**3/(3*sqrt(-1 + x**(-6)
)) - I*acosh(x**(-3))/3 + I/(3*x**3*sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1), (x**3/(3*sqrt(1 - 1/x**6)) + asin(x
**(-3))/3 - 1/(3*x**3*sqrt(1 - 1/x**6)), True))

________________________________________________________________________________________