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

Optimal. Leaf size=38 \[ \frac {7}{12} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1} \left (7 x^3+2\right )}{12 x^6} \]

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {446, 78, 51, 63, 203} \begin {gather*} \frac {7 \sqrt {x^3-1}}{12 x^3}+\frac {7}{12} \tan ^{-1}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^3)/(x^7*Sqrt[-1 + x^3]),x]

[Out]

Sqrt[-1 + x^3]/(6*x^6) + (7*Sqrt[-1 + x^3])/(12*x^3) + (7*ArcTan[Sqrt[-1 + x^3]])/12

Rule 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1+x^3}{x^7 \sqrt {-1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1+x}{\sqrt {-1+x} x^3} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{12} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=\frac {\sqrt {-1+x^3}}{6 x^6}+\frac {7 \sqrt {-1+x^3}}{12 x^3}+\frac {7}{12} \tan ^{-1}\left (\sqrt {-1+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 1.32 \begin {gather*} \frac {1}{12} \sqrt {x^3-1} \left (\frac {7 \tanh ^{-1}\left (\sqrt {1-x^3}\right )}{\sqrt {1-x^3}}+\frac {7 x^3+2}{x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^3]*((2 + 7*x^3)/x^6 + (7*ArcTanh[Sqrt[1 - x^3]])/Sqrt[1 - x^3]))/12

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IntegrateAlgebraic [A]  time = 0.04, size = 38, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^3} \left (2+7 x^3\right )}{12 x^6}+\frac {7}{12} \tan ^{-1}\left (\sqrt {-1+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^3]*(2 + 7*x^3))/(12*x^6) + (7*ArcTan[Sqrt[-1 + x^3]])/12

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fricas [A]  time = 0.46, size = 34, normalized size = 0.89 \begin {gather*} \frac {7 \, x^{6} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (7 \, x^{3} + 2\right )} \sqrt {x^{3} - 1}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(7*x^6*arctan(sqrt(x^3 - 1)) + (7*x^3 + 2)*sqrt(x^3 - 1))/x^6

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giac [A]  time = 0.31, size = 35, normalized size = 0.92 \begin {gather*} \frac {7 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 9 \, \sqrt {x^{3} - 1}}{12 \, x^{6}} + \frac {7}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(7*(x^3 - 1)^(3/2) + 9*sqrt(x^3 - 1))/x^6 + 7/12*arctan(sqrt(x^3 - 1))

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maple [A]  time = 0.34, size = 36, normalized size = 0.95

method	result	size
default	\(\frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {7 \sqrt {x^{3}-1}}{12 x^{3}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(36\)
risch	\(\frac {7 x^{6}-5 x^{3}-2}{12 x^{6} \sqrt {x^{3}-1}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(36\)
elliptic	\(\frac {\sqrt {x^{3}-1}}{6 x^{6}}+\frac {7 \sqrt {x^{3}-1}}{12 x^{3}}+\frac {7 \arctan \left (\sqrt {x^{3}-1}\right )}{12}\)	\(36\)
trager	\(\frac {\sqrt {x^{3}-1}\, \left (7 x^{3}+2\right )}{12 x^{6}}-\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{24}\)	\(63\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }}{x^{3}}-\frac {\left (1-2 \ln \relax (2)+3 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{8 x^{3}}+\frac {\sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )\right )}{3 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \sqrt {\pi }}+\frac {\sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \left (-\frac {\sqrt {\pi }}{2 x^{6}}-\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \relax (2)+3 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}+8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (12 x^{3}+8\right ) \sqrt {-x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{4}\right )}{3 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \sqrt {\pi }}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(x^3-1)^(1/2)/x^6+7/12*(x^3-1)^(1/2)/x^3+7/12*arctan((x^3-1)^(1/2))

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maxima [A]  time = 0.43, size = 60, normalized size = 1.58 \begin {gather*} \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{3} - 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {7}{12} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*(x^3 - 1)^(3/2) + 5*sqrt(x^3 - 1))/(2*x^3 + (x^3 - 1)^2 - 1) + 1/3*sqrt(x^3 - 1)/x^3 + 7/12*arctan(sqr
t(x^3 - 1))

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mupad [B]  time = 0.06, size = 189, normalized size = 4.97 \begin {gather*} \frac {7\,\sqrt {x^3-1}}{12\,x^3}+\frac {\sqrt {x^3-1}}{6\,x^6}-\frac {7\,\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*(x^3 - 1)^(1/2))/(12*x^3) + (x^3 - 1)^(1/2)/(6*x^6) - (7*((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))

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sympy [A]  time = 45.16, size = 63, normalized size = 1.66 \begin {gather*} - \frac {1 - \frac {1}{x^{3} - 1}}{12 \left (1 + \frac {1}{x^{3} - 1}\right )^{2} \sqrt {x^{3} - 1}} - \frac {7 \operatorname {atan}{\left (\frac {1}{\sqrt {x^{3} - 1}} \right )}}{12} + \frac {2}{3 \left (1 + \frac {1}{x^{3} - 1}\right ) \sqrt {x^{3} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1 - 1/(x**3 - 1))/(12*(1 + 1/(x**3 - 1))**2*sqrt(x**3 - 1)) - 7*atan(1/sqrt(x**3 - 1))/12 + 2/(3*(1 + 1/(x**
3 - 1))*sqrt(x**3 - 1))

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