3.16.36 \(\int \frac {1}{x^2 \sqrt {1+x^8}} \, dx\) [1536]

Optimal. Leaf size=37 \[ -\frac {\sqrt {1+x^8}}{x}+\frac {3}{7} x^7 \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^8\right ) \]

[Out]

3/7*x^7*hypergeom([1/2, 7/8],[15/8],-x^8)-(x^8+1)^(1/2)/x

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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 371} \begin {gather*} \frac {3}{7} x^7 \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^8\right )-\frac {\sqrt {x^8+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[1 + x^8]),x]

[Out]

-(Sqrt[1 + x^8]/x) + (3*x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

Rule 331

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt {1+x^8}} \, dx &=-\frac {\sqrt {1+x^8}}{x}+3 \int \frac {x^6}{\sqrt {1+x^8}} \, dx\\ &=-\frac {\sqrt {1+x^8}}{x}+\frac {3}{7} x^7 \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^8\right )\\ \end {align*}

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Mathematica [A]
time = 10.01, size = 20, normalized size = 0.54 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{8},\frac {1}{2};\frac {7}{8};-x^8\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[1 + x^8]),x]

[Out]

-(Hypergeometric2F1[-1/8, 1/2, 7/8, -x^8]/x)

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Maple [A]
time = 0.20, size = 17, normalized size = 0.46

method	result	size
meijerg	\(-\frac {\hypergeom \left (\left [-\frac {1}{8}, \frac {1}{2}\right ], \left [\frac {7}{8}\right ], -x^{8}\right )}{x}\)	\(17\)
risch	\(\frac {3 x^{7} \hypergeom \left (\left [\frac {1}{2}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], -x^{8}\right )}{7}-\frac {\sqrt {x^{8}+1}}{x}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/x*hypergeom([-1/8,1/2],[7/8],-x^8)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^8 + 1)*x^2), x)

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Fricas [F]
time = 0.38, size = 19, normalized size = 0.51 \begin {gather*} {\rm integral}\left (\frac {\sqrt {x^{8} + 1}}{x^{10} + x^{2}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^8 + 1)/(x^10 + x^2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.33, size = 31, normalized size = 0.84 \begin {gather*} \frac {\Gamma \left (- \frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{8}, \frac {1}{2} \\ \frac {7}{8} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x \Gamma \left (\frac {7}{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1/8)*hyper((-1/8, 1/2), (7/8,), x**8*exp_polar(I*pi))/(8*x*gamma(7/8))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^8 + 1)*x^2), x)

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Mupad [B]
time = 1.12, size = 15, normalized size = 0.41 \begin {gather*} -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{8},\frac {1}{2};\ \frac {7}{8};\ -x^8\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x^8 + 1)^(1/2)),x)

[Out]

-hypergeom([-1/8, 1/2], 7/8, -x^8)/x

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