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

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

[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])

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Rubi [A]  time = 0.0244289, antiderivative size = 62, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.0036193, 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]

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

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Maple [C]  time = 0.015, size = 30, normalized size = 0.5 \begin{align*} -{\frac{1}{6\,{x}^{6}}\sqrt{{x}^{8}+1}}-{\frac{{x}^{2}}{6}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{4}};\,-{x}^{8})}} \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{8} + 1} x^{7}}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{8} + 1}}{x^{15} + x^{7}}, x\right ) \end{align*}

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)

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Sympy [C]  time = 0.905108, size = 32, normalized size = 0.52 \begin{align*} \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 )} \end{align*}

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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{8} + 1} x^{7}}\,{d x} \end{align*}

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)