Integrand size = 11, antiderivative size = 49 \[ \int \frac {x^7}{1+x^{12}} \, dx=-\frac {\arctan \left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log \left (1+x^4\right )+\frac {1}{24} \log \left (1-x^4+x^8\right ) \] Output:
-1/12*ln(x^4+1)+1/24*ln(x^8-x^4+1)-1/12*arctan(1/3*(-2*x^4+1)*3^(1/2))*3^( 1/2)
Leaf count is larger than twice the leaf count of optimal. \(260\) vs. \(2(49)=98\).
Time = 0.06 (sec) , antiderivative size = 260, normalized size of antiderivative = 5.31 \[ \int \frac {x^7}{1+x^{12}} \, dx=\frac {1}{24} \left (2 \sqrt {3} \arctan \left (\frac {1+\sqrt {3}-2 \sqrt {2} x}{1-\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1-\sqrt {3}+2 \sqrt {2} x}{1+\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {-1+\sqrt {3}+2 \sqrt {2} x}{1+\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+\sqrt {3}+2 \sqrt {2} x}{-1+\sqrt {3}}\right )-2 \log \left (1-\sqrt {2} x+x^2\right )-2 \log \left (1+\sqrt {2} x+x^2\right )+\log \left (2+\sqrt {2} x-\sqrt {6} x+2 x^2\right )+\log \left (2+\sqrt {2} \left (-1+\sqrt {3}\right ) x+2 x^2\right )+\log \left (2-\left (\sqrt {2}+\sqrt {6}\right ) x+2 x^2\right )+\log \left (2+\left (\sqrt {2}+\sqrt {6}\right ) x+2 x^2\right )\right ) \] Input:
Integrate[x^7/(1 + x^12),x]
Output:
(2*Sqrt[3]*ArcTan[(1 + Sqrt[3] - 2*Sqrt[2]*x)/(1 - Sqrt[3])] - 2*Sqrt[3]*A rcTan[(1 - Sqrt[3] + 2*Sqrt[2]*x)/(1 + Sqrt[3])] + 2*Sqrt[3]*ArcTan[(-1 + Sqrt[3] + 2*Sqrt[2]*x)/(1 + Sqrt[3])] - 2*Sqrt[3]*ArcTan[(1 + Sqrt[3] + 2* Sqrt[2]*x)/(-1 + Sqrt[3])] - 2*Log[1 - Sqrt[2]*x + x^2] - 2*Log[1 + Sqrt[2 ]*x + x^2] + Log[2 + Sqrt[2]*x - Sqrt[6]*x + 2*x^2] + Log[2 + Sqrt[2]*(-1 + Sqrt[3])*x + 2*x^2] + Log[2 - (Sqrt[2] + Sqrt[6])*x + 2*x^2] + Log[2 + ( Sqrt[2] + Sqrt[6])*x + 2*x^2])/24
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {807, 821, 16, 1142, 25, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {x^7}{x^{12}+1} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{4} \int \frac {x^4}{x^{12}+1}dx^4\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {x^4+1}{x^8-x^4+1}dx^4-\frac {1}{3} \int \frac {1}{x^4+1}dx^4\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {x^4+1}{x^8-x^4+1}dx^4-\frac {1}{3} \log \left (x^4+1\right )\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^8-x^4+1}dx^4+\frac {1}{2} \int -\frac {1-2 x^4}{x^8-x^4+1}dx^4\right )-\frac {1}{3} \log \left (x^4+1\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^8-x^4+1}dx^4-\frac {1}{2} \int \frac {1-2 x^4}{x^8-x^4+1}dx^4\right )-\frac {1}{3} \log \left (x^4+1\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (-3 \int \frac {1}{-x^8-3}d\left (2 x^4-1\right )-\frac {1}{2} \int \frac {1-2 x^4}{x^8-x^4+1}dx^4\right )-\frac {1}{3} \log \left (x^4+1\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x^4-1}{\sqrt {3}}\right )-\frac {1}{2} \int \frac {1-2 x^4}{x^8-x^4+1}dx^4\right )-\frac {1}{3} \log \left (x^4+1\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x^4-1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (x^8-x^4+1\right )\right )-\frac {1}{3} \log \left (x^4+1\right )\right )\) |
Input:
Int[x^7/(1 + x^12),x]
Output:
(-1/3*Log[1 + x^4] + (Sqrt[3]*ArcTan[(-1 + 2*x^4)/Sqrt[3]] + Log[1 - x^4 + x^8]/2)/3)/4
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\ln \left (x^{4}+1\right )}{12}+\frac {\ln \left (x^{8}-x^{4}+1\right )}{24}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{4}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}\) | \(39\) |
default | \(-\frac {\ln \left (x^{4}+1\right )}{12}+\frac {\ln \left (x^{8}-x^{4}+1\right )}{24}+\frac {\arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}\) | \(41\) |
meijerg | \(-\frac {x^{8} \ln \left (1+\left (x^{12}\right )^{\frac {1}{3}}\right )}{12 \left (x^{12}\right )^{\frac {2}{3}}}+\frac {x^{8} \ln \left (1-\left (x^{12}\right )^{\frac {1}{3}}+\left (x^{12}\right )^{\frac {2}{3}}\right )}{24 \left (x^{12}\right )^{\frac {2}{3}}}+\frac {x^{8} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{12}\right )^{\frac {1}{3}}}{2-\left (x^{12}\right )^{\frac {1}{3}}}\right )}{12 \left (x^{12}\right )^{\frac {2}{3}}}\) | \(80\) |
Input:
int(x^7/(x^12+1),x,method=_RETURNVERBOSE)
Output:
-1/12*ln(x^4+1)+1/24*ln(x^8-x^4+1)+1/12*3^(1/2)*arctan(2/3*(x^4-1/2)*3^(1/ 2))
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1+x^{12}} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \] Input:
integrate(x^7/(x^12+1),x, algorithm="fricas")
Output:
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) + 1/24*log(x^8 - x^4 + 1) - 1 /12*log(x^4 + 1)
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {x^7}{1+x^{12}} \, dx=- \frac {\log {\left (x^{4} + 1 \right )}}{12} + \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{24} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \] Input:
integrate(x**7/(x**12+1),x)
Output:
-log(x**4 + 1)/12 + log(x**8 - x**4 + 1)/24 + sqrt(3)*atan(2*sqrt(3)*x**4/ 3 - sqrt(3)/3)/12
Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1+x^{12}} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \] Input:
integrate(x^7/(x^12+1),x, algorithm="maxima")
Output:
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) + 1/24*log(x^8 - x^4 + 1) - 1 /12*log(x^4 + 1)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1+x^{12}} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \] Input:
integrate(x^7/(x^12+1),x, algorithm="giac")
Output:
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) + 1/24*log(x^8 - x^4 + 1) - 1 /12*log(x^4 + 1)
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {x^7}{1+x^{12}} \, dx=-\frac {\ln \left (x^4+1\right )}{12}-\ln \left (x^4-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (x^4+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right ) \] Input:
int(x^7/(x^12 + 1),x)
Output:
log((3^(1/2)*1i)/2 + x^4 - 1/2)*((3^(1/2)*1i)/24 + 1/24) - log(x^4 - (3^(1 /2)*1i)/2 - 1/2)*((3^(1/2)*1i)/24 - 1/24) - log(x^4 + 1)/12
Time = 0.17 (sec) , antiderivative size = 353, normalized size of antiderivative = 7.20 \[ \int \frac {x^7}{1+x^{12}} \, dx=-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{24}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{24}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}+\frac {\mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}-\frac {\mathrm {log}\left (-\sqrt {2}\, x +x^{2}+1\right )}{12}+\frac {\mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}-\frac {\mathrm {log}\left (\sqrt {2}\, x +x^{2}+1\right )}{12}+\frac {\mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{24}+\frac {\mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{24} \] Input:
int(x^7/(x^12+1),x)
Output:
( - sqrt( - sqrt(3) + 2)*sqrt(6)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) - sqrt( - sqrt(3) + 2)*sqrt(6)*atan((sqrt( 6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2)*sqr t(2)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) + sqrt( - sq rt(3) + 2)*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2)) ) + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sq rt(6) + sqrt(2))) + sqrt( - sqrt(3) + 2)*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*s qrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) + log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1) - 2*log( - sqrt(2)*x + x**2 + 1) + log(sqrt( - sqrt(3) + 2)*x + x**2 + 1) - 2*log(sqrt(2)*x + x**2 + 1) + log(( - sqrt(6)*x - sqrt (2)*x + 2*x**2 + 2)/2) + log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2))/24