3.5.14 \(\int \frac {-1+x^{16}}{x^8 \sqrt {-1+x^4}} \, dx\)

Optimal. Leaf size=33 \[ \frac {\sqrt {x^4-1} \left (3 x^{12}+5 x^8-5 x^4-3\right )}{21 x^7} \]

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Rubi [A]  time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.42, 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 = {1586, 1835, 1585, 1486, 449} \begin {gather*} \frac {1}{7} x \left (x^4-1\right )^{3/2}+\frac {\left (x^4-1\right )^{3/2}}{7 x^7}+\frac {8 \left (x^4-1\right )^{3/2}}{21 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^16)/(x^8*Sqrt[-1 + x^4]),x]

[Out]

(-1 + x^4)^(3/2)/(7*x^7) + (8*(-1 + x^4)^(3/2))/(21*x^3) + (x*(-1 + x^4)^(3/2))/7

Rule 449

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rule 1486

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Simp[(c^p*(f*x)^(m + 2*n*p - n + 1)*(d + e*x^n)^(q + 1))/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1)),
x] + Dist[1/(e*(m + 2*n*p + n*q + 1)), Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e*(m + 2*n*p + n*q + 1)*((a + b*
x^n + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x], x] /; FreeQ[{a, b, c, d
, e, f, m, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] &&
!IntegerQ[q] && NeQ[m + 2*n*p + n*q + 1, 0]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1835

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[(Pq
0*(c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[(2*a*(m + 1)*(Pq - Pq0))/x - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rubi steps

\begin {align*} \int \frac {-1+x^{16}}{x^8 \sqrt {-1+x^4}} \, dx &=\int \frac {\sqrt {-1+x^4} \left (1+x^4+x^8+x^{12}\right )}{x^8} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/2}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt {-1+x^4} \left (16 x^3+14 x^7+14 x^{11}\right )}{x^7} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/2}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt {-1+x^4} \left (16+14 x^4+14 x^8\right )}{x^4} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/2}}{7 x^7}+\frac {1}{7} x \left (-1+x^4\right )^{3/2}+\frac {1}{98} \int \frac {\sqrt {-1+x^4} \left (112+112 x^4\right )}{x^4} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/2}}{7 x^7}+\frac {8 \left (-1+x^4\right )^{3/2}}{21 x^3}+\frac {1}{7} x \left (-1+x^4\right )^{3/2}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 85, normalized size = 2.58 \begin {gather*} \frac {3 \sqrt {1-x^4} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};x^4\right )+x^8 \left (5 \sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )+3 x^8+2 x^4-5\right )}{21 x^7 \sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^16)/(x^8*Sqrt[-1 + x^4]),x]

[Out]

(3*Sqrt[1 - x^4]*Hypergeometric2F1[-7/4, 1/2, -3/4, x^4] + x^8*(-5 + 2*x^4 + 3*x^8 + 5*Sqrt[1 - x^4]*Hypergeom
etric2F1[1/4, 1/2, 5/4, x^4]))/(21*x^7*Sqrt[-1 + x^4])

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IntegrateAlgebraic [A]  time = 0.20, size = 33, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^4} \left (-3-5 x^4+5 x^8+3 x^{12}\right )}{21 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^16)/(x^8*Sqrt[-1 + x^4]),x]

[Out]

(Sqrt[-1 + x^4]*(-3 - 5*x^4 + 5*x^8 + 3*x^12))/(21*x^7)

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fricas [A]  time = 0.66, size = 29, normalized size = 0.88 \begin {gather*} \frac {{\left (3 \, x^{12} + 5 \, x^{8} - 5 \, x^{4} - 3\right )} \sqrt {x^{4} - 1}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/x^8/(x^4-1)^(1/2),x, algorithm="fricas")

[Out]

1/21*(3*x^12 + 5*x^8 - 5*x^4 - 3)*sqrt(x^4 - 1)/x^7

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giac [A]  time = 0.43, size = 39, normalized size = 1.18 \begin {gather*} \frac {1}{21} \, {\left (3 \, x^{4} + 5\right )} \sqrt {x^{4} - 1} x - \frac {{\left (\frac {3}{x^{4}} + 5\right )} \sqrt {-\frac {1}{x^{4}} + 1}}{21 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/x^8/(x^4-1)^(1/2),x, algorithm="giac")

[Out]

1/21*(3*x^4 + 5)*sqrt(x^4 - 1)*x - 1/21*(3/x^4 + 5)*sqrt(-1/x^4 + 1)/x

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maple [A]  time = 0.13, size = 30, normalized size = 0.91

method	result	size
trager	\(\frac {\sqrt {x^{4}-1}\, \left (3 x^{12}+5 x^{8}-5 x^{4}-3\right )}{21 x^{7}}\)	\(30\)
risch	\(\frac {3 x^{16}+2 x^{12}-10 x^{8}+2 x^{4}+3}{21 x^{7} \sqrt {x^{4}-1}}\)	\(35\)
gosper	\(\frac {\left (x^{2}+1\right ) \left (1+x \right ) \left (-1+x \right ) \left (3 x^{8}+8 x^{4}+3\right ) \sqrt {x^{4}-1}}{21 x^{7}}\)	\(36\)
elliptic	\(\frac {\left (\frac {\left (x^{4}-1\right )^{\frac {7}{2}} \sqrt {2}}{7 x^{7}}+\frac {2 \left (x^{4}-1\right )^{\frac {3}{2}} \sqrt {2}}{3 x^{3}}\right ) \sqrt {2}}{2}\)	\(37\)
default	\(\frac {x^{5} \sqrt {x^{4}-1}}{7}+\frac {5 x \sqrt {x^{4}-1}}{21}-\frac {\sqrt {x^{4}-1}}{7 x^{7}}-\frac {5 \sqrt {x^{4}-1}}{21 x^{3}}\)	\(48\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \hypergeom \left (\left [\frac {1}{2}, \frac {9}{4}\right ], \left [\frac {13}{4}\right ], x^{4}\right ) x^{9}}{9 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \hypergeom \left (\left [-\frac {7}{4}, \frac {1}{2}\right ], \left [-\frac {3}{4}\right ], x^{4}\right )}{7 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}\, x^{7}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^16-1)/x^8/(x^4-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/21*(x^4-1)^(1/2)*(3*x^12+5*x^8-5*x^4-3)/x^7

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maxima [A]  time = 0.53, size = 39, normalized size = 1.18 \begin {gather*} \frac {{\left (3 \, x^{12} + 5 \, x^{8} - 5 \, x^{4} - 3\right )} \sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/x^8/(x^4-1)^(1/2),x, algorithm="maxima")

[Out]

1/21*(3*x^12 + 5*x^8 - 5*x^4 - 3)*sqrt(x^2 + 1)*sqrt(x + 1)*sqrt(x - 1)/x^7

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mupad [B]  time = 0.34, size = 29, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {x^4-1}\,\left (-3\,x^{12}-5\,x^8+5\,x^4+3\right )}{21\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^16 - 1)/(x^8*(x^4 - 1)^(1/2)),x)

[Out]

-((x^4 - 1)^(1/2)*(5*x^4 - 5*x^8 - 3*x^12 + 3))/(21*x^7)

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sympy [C]  time = 2.73, size = 60, normalized size = 1.82 \begin {gather*} - \frac {i x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {i \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**16-1)/x**8/(x**4-1)**(1/2),x)

[Out]

-I*x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), x**4)/(4*gamma(13/4)) + I*gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,)
, x**4)/(4*x**7*gamma(-3/4))

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