Integrand size = 22, antiderivative size = 52 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {\text {arctanh}\left (\frac {\sqrt {1+x^{16}}}{\sqrt {2}}\right )}{4 \sqrt {2}} \] Output:
-1/8*(x^16+1)^(1/2)-1/24*(x^16+1)^(3/2)+1/8*arctanh(1/2*(x^16+1)^(1/2)*2^( 1/2))*2^(1/2)
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=\frac {1}{24} \left (-4-x^{16}\right ) \sqrt {1+x^{16}}+\frac {\text {arctanh}\left (\frac {\sqrt {1+x^{16}}}{\sqrt {2}}\right )}{4 \sqrt {2}} \] Input:
Integrate[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]
Output:
((-4 - x^16)*Sqrt[1 + x^16])/24 + ArcTanh[Sqrt[1 + x^16]/Sqrt[2]]/(4*Sqrt[ 2])
Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 90, 60, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^{31} \sqrt {x^{16}+1}}{1-x^{16}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{16} \int \frac {x^{16} \sqrt {x^{16}+1}}{1-x^{16}}dx^{16}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{16} \left (\int \frac {\sqrt {x^{16}+1}}{1-x^{16}}dx^{16}-\frac {2}{3} \left (x^{16}+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{16} \left (2 \int \frac {1}{\left (1-x^{16}\right ) \sqrt {x^{16}+1}}dx^{16}-\frac {2}{3} \left (x^{16}+1\right )^{3/2}-2 \sqrt {x^{16}+1}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{16} \left (4 \int \frac {1}{2-x^{32}}d\sqrt {x^{16}+1}-\frac {2}{3} \left (x^{16}+1\right )^{3/2}-2 \sqrt {x^{16}+1}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{16} \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )-\frac {2}{3} \left (x^{16}+1\right )^{3/2}-2 \sqrt {x^{16}+1}\right )\) |
Input:
Int[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]
Output:
(-2*Sqrt[1 + x^16] - (2*(1 + x^16)^(3/2))/3 + 2*Sqrt[2]*ArcTanh[Sqrt[1 + x ^16]/Sqrt[2]])/16
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(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]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[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]]
Time = 0.69 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.33
method | result | size |
pseudoelliptic | \(-\frac {x^{16} \sqrt {x^{16}+1}}{24}-\frac {\sqrt {x^{16}+1}}{6}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{8}-1\right ) \sqrt {2}}{2 \sqrt {x^{16}+1}}\right )}{16}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{8}+1\right ) \sqrt {2}}{2 \sqrt {x^{16}+1}}\right )}{16}\) | \(69\) |
trager | \(\left (-\frac {x^{16}}{24}-\frac {1}{6}\right ) \sqrt {x^{16}+1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{16}+4 \sqrt {x^{16}+1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}+1\right )}\right )}{16}\) | \(87\) |
risch | \(-\frac {\left (x^{16}+4\right ) \sqrt {x^{16}+1}}{24}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{16}+4 \sqrt {x^{16}+1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}+1\right )}\right )}{16}\) | \(87\) |
Input:
int(x^31*(x^16+1)^(1/2)/(-x^16+1),x,method=_RETURNVERBOSE)
Output:
-1/24*x^16*(x^16+1)^(1/2)-1/6*(x^16+1)^(1/2)-1/16*2^(1/2)*arctanh(1/2*(x^8 -1)*2^(1/2)/(x^16+1)^(1/2))+1/16*2^(1/2)*arctanh(1/2*(x^8+1)*2^(1/2)/(x^16 +1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=-\frac {1}{24} \, {\left (x^{16} + 4\right )} \sqrt {x^{16} + 1} + \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{16} + 2 \, \sqrt {2} \sqrt {x^{16} + 1} + 3}{x^{16} - 1}\right ) \] Input:
integrate(x^31*(x^16+1)^(1/2)/(-x^16+1),x, algorithm="fricas")
Output:
-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))
Time = 34.77 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.12 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=- \frac {\left (x^{16} + 1\right )^{\frac {3}{2}}}{24} - \frac {\sqrt {x^{16} + 1}}{8} - \frac {\sqrt {2} \left (\log {\left (\sqrt {x^{16} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {x^{16} + 1} + \sqrt {2} \right )}\right )}{16} \] Input:
integrate(x**31*(x**16+1)**(1/2)/(-x**16+1),x)
Output:
-(x**16 + 1)**(3/2)/24 - sqrt(x**16 + 1)/8 - sqrt(2)*(log(sqrt(x**16 + 1) - sqrt(2)) - log(sqrt(x**16 + 1) + sqrt(2)))/16
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.02 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=-\frac {1}{24} \, {\left (x^{16} + 1\right )}^{\frac {3}{2}} - \frac {1}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{16} + 1}}{\sqrt {2} + \sqrt {x^{16} + 1}}\right ) - \frac {1}{8} \, \sqrt {x^{16} + 1} \] Input:
integrate(x^31*(x^16+1)^(1/2)/(-x^16+1),x, algorithm="maxima")
Output:
-1/24*(x^16 + 1)^(3/2) - 1/16*sqrt(2)*log(-(sqrt(2) - sqrt(x^16 + 1))/(sqr t(2) + sqrt(x^16 + 1))) - 1/8*sqrt(x^16 + 1)
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.08 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=-\frac {1}{24} \, {\left (x^{16} + 1\right )}^{\frac {3}{2}} - \frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{16} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{16} + 1}\right )}}\right ) - \frac {1}{8} \, \sqrt {x^{16} + 1} \] Input:
integrate(x^31*(x^16+1)^(1/2)/(-x^16+1),x, algorithm="giac")
Output:
-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)
Time = 3.51 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71 \[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^{16}+1}}{2}\right )}{8}-\frac {\sqrt {x^{16}+1}}{8}-\frac {{\left (x^{16}+1\right )}^{3/2}}{24} \] Input:
int(-(x^31*(x^16 + 1)^(1/2))/(x^16 - 1),x)
Output:
(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
\[ \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx=-\frac {\sqrt {x^{16}+1}\, x^{16}}{24}+\frac {\sqrt {x^{16}+1}}{12}-2 \left (\int \frac {\sqrt {x^{16}+1}\, x^{31}}{x^{32}-1}d x \right ) \] Input:
int(x^31*(x^16+1)^(1/2)/(-x^16+1),x)
Output:
( - sqrt(x**16 + 1)*x**16 + 2*sqrt(x**16 + 1) - 48*int((sqrt(x**16 + 1)*x* *31)/(x**32 - 1),x))/24