3.1064 \(\int \frac{x^{31} \sqrt{1+x^{16}}}{1-x^{16}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.027784, antiderivative size = 52, normalized size of antiderivative = 1., 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} \[ -\frac{1}{24} \left (x^{16}+1\right )^{3/2}-\frac{\sqrt{x^{16}+1}}{8}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

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 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]]

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 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 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])

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*}

Mathematica [A]  time = 0.0182098, size = 44, normalized size = 0.85 \[ \frac{1}{24} \left (3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right )-\sqrt{x^{16}+1} \left (x^{16}+4\right )\right ) \]

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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Maple [F]  time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{31}}{-{x}^{16}+1}\sqrt{{x}^{16}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42199, size = 72, normalized size = 1.38 \begin{align*} -\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{align*}

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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Fricas [A]  time = 0.997523, size = 138, normalized size = 2.65 \begin{align*} -\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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.07295, size = 76, normalized size = 1.46 \begin{align*} -\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{align*}

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)