3.11.64 \(\int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx\) [1064]

Optimal. Leaf size=52 \[ -\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {1+x^{16}}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]

[Out]

-1/24*(x^16+1)^(3/2)+1/8*arctanh(1/2*(x^16+1)^(1/2)*2^(1/2))*2^(1/2)-1/8*(x^16+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 81, 52, 65, 212} \begin {gather*} -\frac {1}{24} \left (x^{16}+1\right )^{3/2}-\frac {\sqrt {x^{16}+1}}{8}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]

[Out]

-1/8*Sqrt[1 + x^16] - (1 + x^16)^(3/2)/24 + ArcTanh[Sqrt[1 + x^16]/Sqrt[2]]/(4*Sqrt[2])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx &=\frac {1}{16} \text {Subst}\left (\int \frac {x \sqrt {1+x}}{1-x} \, dx,x,x^{16}\right )\\ &=-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {1}{16} \text {Subst}\left (\int \frac {\sqrt {1+x}}{1-x} \, dx,x,x^{16}\right )\\ &=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,x^{16}\right )\\ &=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+x^{16}}\right )\\ &=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {1+x^{16}}}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 46, normalized size = 0.88 \begin {gather*} \frac {1}{24} \left (-4-x^{16}\right ) \sqrt {1+x^{16}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {1+x^{16}}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]

[Out]

((-4 - x^16)*Sqrt[1 + x^16])/24 + ArcTanh[Sqrt[1 + x^16]/Sqrt[2]]/(4*Sqrt[2])

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.69, size = 85, normalized size = 1.63

method	result	size
risch	\(-\frac {\left (x^{16}+4\right ) \sqrt {x^{16}+1}}{24}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{16}+3 \RootOf \left (\textit {\_Z}^{2}-2\right )-4 \sqrt {x^{16}+1}}{\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}+1\right )}\right )}{16}\)	\(85\)
trager	\(\left (-\frac {x^{16}}{24}-\frac {1}{6}\right ) \sqrt {x^{16}+1}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{16}+3 \RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x^{16}+1}}{\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}+1\right )}\right )}{16}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^31*(x^16+1)^(1/2)/(-x^16+1),x,method=_RETURNVERBOSE)

[Out]

-1/24*(x^16+4)*(x^16+1)^(1/2)-1/16*RootOf(_Z^2-2)*ln((RootOf(_Z^2-2)*x^16+3*RootOf(_Z^2-2)-4*(x^16+1)^(1/2))/(
x-1)/(x+1)/(x^2+1)/(x^4+1)/(x^8+1))

________________________________________________________________________________________

Maxima [A]
time = 0.56, size = 53, normalized size = 1.02 \begin {gather*} -\frac {1}{24} \, {\left (x^{16} + 1\right )}^{\frac {3}{2}} - \frac {1}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{16} + 1}}{\sqrt {2} + \sqrt {x^{16} + 1}}\right ) - \frac {1}{8} \, \sqrt {x^{16} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^31*(x^16+1)^(1/2)/(-x^16+1),x, algorithm="maxima")

[Out]

-1/24*(x^16 + 1)^(3/2) - 1/16*sqrt(2)*log(-(sqrt(2) - sqrt(x^16 + 1))/(sqrt(2) + sqrt(x^16 + 1))) - 1/8*sqrt(x
^16 + 1)

________________________________________________________________________________________

Fricas [A]
time = 2.46, size = 46, normalized size = 0.88 \begin {gather*} -\frac {1}{24} \, {\left (x^{16} + 4\right )} \sqrt {x^{16} + 1} + \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{16} + 2 \, \sqrt {2} \sqrt {x^{16} + 1} + 3}{x^{16} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^31*(x^16+1)^(1/2)/(-x^16+1),x, algorithm="fricas")

[Out]

-1/24*(x^16 + 4)*sqrt(x^16 + 1) + 1/16*sqrt(2)*log((x^16 + 2*sqrt(2)*sqrt(x^16 + 1) + 3)/(x^16 - 1))

________________________________________________________________________________________

Sympy [A]
time = 112.71, size = 76, normalized size = 1.46 \begin {gather*} - \frac {\left (x^{16} + 1\right )^{\frac {3}{2}}}{24} - \frac {\sqrt {x^{16} + 1}}{8} - \frac {\begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {x^{16} + 1}}{2} \right )}}{2} & \text {for}\: x^{16} > 1 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {x^{16} + 1}}{2} \right )}}{2} & \text {for}\: x^{16} < 1 \end {cases}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**31*(x**16+1)**(1/2)/(-x**16+1),x)

[Out]

-(x**16 + 1)**(3/2)/24 - sqrt(x**16 + 1)/8 - Piecewise((-sqrt(2)*acoth(sqrt(2)*sqrt(x**16 + 1)/2)/2, x**16 > 1
), (-sqrt(2)*atanh(sqrt(2)*sqrt(x**16 + 1)/2)/2, x**16 < 1))/4

________________________________________________________________________________________

Giac [A]
time = 1.15, size = 56, normalized size = 1.08 \begin {gather*} -\frac {1}{24} \, {\left (x^{16} + 1\right )}^{\frac {3}{2}} - \frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{16} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{16} + 1}\right )}}\right ) - \frac {1}{8} \, \sqrt {x^{16} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^31*(x^16+1)^(1/2)/(-x^16+1),x, algorithm="giac")

[Out]

-1/24*(x^16 + 1)^(3/2) - 1/16*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(x^16 + 1))/(sqrt(2) + sqrt(x^16 + 1))) -
1/8*sqrt(x^16 + 1)

________________________________________________________________________________________

Mupad [B]
time = 4.82, size = 37, normalized size = 0.71 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^{16}+1}}{2}\right )}{8}-\frac {\sqrt {x^{16}+1}}{8}-\frac {{\left (x^{16}+1\right )}^{3/2}}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^31*(x^16 + 1)^(1/2))/(x^16 - 1),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(x^16 + 1)^(1/2))/2))/8 - (x^16 + 1)^(1/2)/8 - (x^16 + 1)^(3/2)/24

________________________________________________________________________________________