3.1529 \(\int \frac {1}{x^7 \sqrt {1+x^8}} \, dx\)

Optimal. Leaf size=62 \[ -\frac {\sqrt {x^8+1}}{6 x^6}-\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{12 \sqrt {x^8+1}} \]

[Out]

-1/6*(x^8+1)^(1/2)/x^6-1/12*(x^4+1)*(cos(2*arctan(x^2))^2)^(1/2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2
)),1/2*2^(1/2))*((x^8+1)/(x^4+1)^2)^(1/2)/(x^8+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 325, 220} \[ -\frac {\sqrt {x^8+1}}{6 x^6}-\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{12 \sqrt {x^8+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(6*x^6) - ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(12*Sqrt[1 + x^
8])

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{x^7 \sqrt {1+x^8}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+x^8}}{6 x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+x^8}}{6 x^6}-\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{12 \sqrt {1+x^8}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 22, normalized size = 0.35 \[ -\frac {\, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-x^8\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[1 + x^8]),x]

[Out]

-1/6*Hypergeometric2F1[-3/4, 1/2, 1/4, -x^8]/x^6

________________________________________________________________________________________

fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{8} + 1}}{x^{15} + x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^8 + 1)/(x^15 + x^7), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{8} + 1} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^7), x)

________________________________________________________________________________________

maple [C]  time = 0.02, size = 30, normalized size = 0.48 \[ -\frac {x^{2} \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -x^{8}\right )}{6}-\frac {\sqrt {x^{8}+1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^8+1)^(1/2),x)

[Out]

-1/6*(x^8+1)^(1/2)/x^6-1/6*x^2*hypergeom([1/4,1/2],[5/4],-x^8)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{8} + 1} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^7), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^7\,\sqrt {x^8+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(x^8 + 1)^(1/2)),x)

[Out]

int(1/(x^7*(x^8 + 1)^(1/2)), x)

________________________________________________________________________________________

sympy [C]  time = 0.99, size = 32, normalized size = 0.52 \[ \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x^{6} \Gamma \left (\frac {1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**8+1)**(1/2),x)

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**8*exp_polar(I*pi))/(8*x**6*gamma(1/4))

________________________________________________________________________________________