3.7.85 \(\int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx\)

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

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Rubi [A]  time = 0.03, 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 = {446, 80, 50, 63, 206} \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]

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

Rule 50

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 63

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 80

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 206

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

Rule 446

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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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*}

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Mathematica [A]  time = 0.04, size = 44, normalized size = 0.85 \begin {gather*} \frac {1}{24} \left (3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )-\sqrt {x^{16}+1} \left (x^{16}+4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.07, size = 46, normalized size = 0.88 \begin {gather*} \frac {1}{24} \sqrt {x^{16}+1} \left (-x^{16}-4\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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])

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fricas [A]  time = 0.42, 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))

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giac [A]  time = 0.15, 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)

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maple [C]  time = 1.15, size = 86, normalized size = 1.65 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {x^{16} \RootOf \left (\textit {\_Z}^{2}-2\right )+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}-\frac {\left (x^{16}+4\right ) \sqrt {x^{16}+1}}{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]

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

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maxima [A]  time = 1.25, 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)

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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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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