\(\int \frac {1}{x^7 \sqrt {1+x^3}} \, dx\) [444]

Optimal result

Integrand size = 13, antiderivative size = 47 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {\sqrt {1+x^3}}{6 x^6}+\frac {\sqrt {1+x^3}}{4 x^3}-\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65, 213} \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {1}{4} \text {arctanh}\left (\sqrt {x^3+1}\right )+\frac {\sqrt {x^3+1}}{4 x^3}-\frac {\sqrt {x^3+1}}{6 x^6} \]

[In]

Int[1/(x^7*Sqrt[1 + x^3]),x]

[Out]

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

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[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 {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1+x^3}}{6 x^6}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1+x^3}}{6 x^6}+\frac {\sqrt {1+x^3}}{4 x^3}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {1+x^3}}{6 x^6}+\frac {\sqrt {1+x^3}}{4 x^3}+\frac {1}{4} \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}}{4 x^3}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {\sqrt {1+x^3} \left (-2+3 x^3\right )}{12 x^6}-\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^3}\right ) \]

[In]

Integrate[1/(x^7*Sqrt[1 + x^3]),x]

[Out]

(Sqrt[1 + x^3]*(-2 + 3*x^3))/(12*x^6) - ArcTanh[Sqrt[1 + x^3]]/4

Maple [A] (verified)

Time = 4.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72

method	result	size
risch	\(\frac {3 x^{6}+x^{3}-2}{12 x^{6} \sqrt {x^{3}+1}}-\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{4}\)	\(34\)
default	\(-\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{4}-\frac {\sqrt {x^{3}+1}}{6 x^{6}}+\frac {\sqrt {x^{3}+1}}{4 x^{3}}\)	\(36\)
elliptic	\(-\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{4}-\frac {\sqrt {x^{3}+1}}{6 x^{6}}+\frac {\sqrt {x^{3}+1}}{4 x^{3}}\)	\(36\)
trager	\(\frac {\left (3 x^{3}-2\right ) \sqrt {x^{3}+1}}{12 x^{6}}-\frac {\ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{8}\)	\(43\)
pseudoelliptic	\(\frac {3 \ln \left (-1+\sqrt {x^{3}+1}\right ) x^{6}-3 \ln \left (1+\sqrt {x^{3}+1}\right ) x^{6}+6 x^{3} \sqrt {x^{3}+1}-4 \sqrt {x^{3}+1}}{24 \left (-1+\sqrt {x^{3}+1}\right )^{2} \left (1+\sqrt {x^{3}+1}\right )^{2}}\)	\(77\)
meijerg	\(\frac {-\frac {\sqrt {\pi }}{2 x^{6}}+\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}-8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (-12 x^{3}+8\right ) \sqrt {x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )}{4}}{3 \sqrt {\pi }}\)	\(97\)

[In]

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

[Out]

1/12*(3*x^6+x^3-2)/x^6/(x^3+1)^(1/2)-1/4*arctanh((x^3+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {3 \, x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (3 \, x^{3} - 2\right )} \sqrt {x^{3} + 1}}{24 \, x^{6}} \]

[In]

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

[Out]

-1/24*(3*x^6*log(sqrt(x^3 + 1) + 1) - 3*x^6*log(sqrt(x^3 + 1) - 1) - 2*(3*x^3 - 2)*sqrt(x^3 + 1))/x^6

Sympy [A] (verification not implemented)

Time = 2.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=- \frac {\operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} + \frac {1}{4 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {1}{12 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \]

[In]

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

[Out]

-asinh(x**(-3/2))/4 + 1/(4*x**(3/2)*sqrt(1 + x**(-3))) + 1/(12*x**(9/2)*sqrt(1 + x**(-3))) - 1/(6*x**(15/2)*sq
rt(1 + x**(-3)))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{3} + 1}}{12 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} - \frac {1}{8} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]

[In]

integrate(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) - 1/8*log(sqrt(x^3 + 1) + 1) + 1/8*log(s
qrt(x^3 + 1) - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{3} + 1}}{12 \, x^{6}} - \frac {1}{8} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{8} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \]

[In]

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

[Out]

1/12*(3*(x^3 + 1)^(3/2) - 5*sqrt(x^3 + 1))/x^6 - 1/8*log(sqrt(x^3 + 1) + 1) + 1/8*log(abs(sqrt(x^3 + 1) - 1))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.02 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {\sqrt {x^3+1}}{4\,x^3}-\frac {\sqrt {x^3+1}}{6\,x^6}-\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int(1/(x^7*(x^3 + 1)^(1/2)),x)

[Out]

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