Integrand size = 9, antiderivative size = 210 \[ \int \frac {1}{a+b x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x}{\sqrt [4]{-a}+\sqrt [4]{b} x^2}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}} \] Output:
-1/4*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(7/8)/b^(1/8)-1/8*arctan(-1+2^(1/2) *b^(1/8)*x/(-a)^(1/8))*2^(1/2)/(-a)^(7/8)/b^(1/8)-1/8*arctan(1+2^(1/2)*b^( 1/8)*x/(-a)^(1/8))*2^(1/2)/(-a)^(7/8)/b^(1/8)-1/4*arctanh(b^(1/8)*x/(-a)^( 1/8))/(-a)^(7/8)/b^(1/8)-1/8*arctanh(2^(1/2)*(-a)^(1/8)*b^(1/8)*x/((-a)^(1 /4)+b^(1/4)*x^2))*2^(1/2)/(-a)^(7/8)/b^(1/8)
Time = 0.18 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.54 \[ \int \frac {1}{a+b x^8} \, dx=\frac {2 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+2 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-2 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+2 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 a^{7/8} \sqrt [8]{b}} \] Input:
Integrate[(a + b*x^8)^(-1),x]
Output:
(2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] + 2*ArcTan[ (b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - Cos[Pi/8]*Log[a^(1/ 4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - 2*ArcTan[Cot[Pi/8] - (b^( 1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + 2*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc [Pi/8])/a^(1/8)]*Sin[Pi/8] - Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8) *x*Sin[Pi/8]]*Sin[Pi/8] + Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x* Sin[Pi/8]]*Sin[Pi/8])/(8*a^(7/8)*b^(1/8))
Time = 0.87 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.48, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {758, 755, 756, 218, 221, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {1}{a+b x^8} \, dx\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle -\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{b} x^2+\sqrt [4]{-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [8]{-a} \sqrt [4]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\) |
Input:
Int[(a + b*x^8)^(-1),x]
Output:
-1/2*(ArcTan[(b^(1/8)*x)/(-a)^(1/8)]/(2*(-a)^(3/8)*b^(1/8)) + ArcTanh[(b^( 1/8)*x)/(-a)^(1/8)]/(2*(-a)^(3/8)*b^(1/8)))/Sqrt[-a] - ((-(ArcTan[1 - (Sqr t[2]*b^(1/8)*x)/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8))) + ArcTan[1 + (Sq rt[2]*b^(1/8)*x)/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8)))/(2*(-a)^(1/4)) + (-1/2*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2]/(Sqrt [2]*(-a)^(1/8)*b^(1/8)) + Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2]/(2*Sqrt[2]*(-a)^(1/8)*b^(1/8)))/(2*(-a)^(1/4)))/(2*Sqrt[-a])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b , 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/b, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\) | \(27\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\) | \(27\) |
Input:
int(1/(b*x^8+a),x,method=_RETURNVERBOSE)
Output:
1/8/b*sum(1/_R^7*ln(x-_R),_R=RootOf(_Z^8*b+a))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.18 \[ \int \frac {1}{a+b x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} i \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (i \, a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} i \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-i \, a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) \] Input:
integrate(1/(b*x^8+a),x, algorithm="fricas")
Output:
(1/16*I + 1/16)*sqrt(2)*(-1/(a^7*b))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*(-1 /(a^7*b))^(1/8) + x) - (1/16*I - 1/16)*sqrt(2)*(-1/(a^7*b))^(1/8)*log(-(1/ 2*I - 1/2)*sqrt(2)*a*(-1/(a^7*b))^(1/8) + x) + (1/16*I - 1/16)*sqrt(2)*(-1 /(a^7*b))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a*(-1/(a^7*b))^(1/8) + x) - (1/1 6*I + 1/16)*sqrt(2)*(-1/(a^7*b))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a*(-1/(a ^7*b))^(1/8) + x) + 1/8*(-1/(a^7*b))^(1/8)*log(a*(-1/(a^7*b))^(1/8) + x) + 1/8*I*(-1/(a^7*b))^(1/8)*log(I*a*(-1/(a^7*b))^(1/8) + x) - 1/8*I*(-1/(a^7 *b))^(1/8)*log(-I*a*(-1/(a^7*b))^(1/8) + x) - 1/8*(-1/(a^7*b))^(1/8)*log(- a*(-1/(a^7*b))^(1/8) + x)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.10 \[ \int \frac {1}{a+b x^8} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{7} b + 1, \left ( t \mapsto t \log {\left (8 t a + x \right )} \right )\right )} \] Input:
integrate(1/(b*x**8+a),x)
Output:
RootSum(16777216*_t**8*a**7*b + 1, Lambda(_t, _t*log(8*_t*a + x)))
\[ \int \frac {1}{a+b x^8} \, dx=\int { \frac {1}{b x^{8} + a} \,d x } \] Input:
integrate(1/(b*x^8+a),x, algorithm="maxima")
Output:
integrate(1/(b*x^8 + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (143) = 286\).
Time = 0.13 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.08 \[ \int \frac {1}{a+b x^8} \, dx=\frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} \] Input:
integrate(1/(b*x^8+a),x, algorithm="giac")
Output:
1/4*(a/b)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2 ) + 2)*(a/b)^(1/8)))/(a*sqrt(-2*sqrt(2) + 4)) + 1/4*(a/b)^(1/8)*arctan((2* x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/(a*sq rt(-2*sqrt(2) + 4)) + 1/4*(a/b)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b )^(1/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) + 1/4*( a/b)^(1/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2) + 2 )*(a/b)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) + 1/8*(a/b)^(1/8)*log(x^2 + x*sqrt (sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a*sqrt(-2*sqrt(2) + 4)) - 1/8*(a /b)^(1/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a*sqrt (-2*sqrt(2) + 4)) + 1/8*(a/b)^(1/8)*log(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^( 1/8) + (a/b)^(1/4))/(a*sqrt(2*sqrt(2) + 4)) - 1/8*(a/b)^(1/8)*log(x^2 - x* sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a*sqrt(2*sqrt(2) + 4))
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.52 \[ \int \frac {1}{a+b x^8} \, dx=-\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x}{{\left (-a\right )}^{1/8}}\right )}{4\,{\left (-a\right )}^{7/8}\,b^{1/8}}+\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,b^{1/8}} \] Input:
int(1/(a + b*x^8),x)
Output:
(atan((b^(1/8)*x*1i)/(-a)^(1/8))*1i)/(4*(-a)^(7/8)*b^(1/8)) - atan((b^(1/8 )*x)/(-a)^(1/8))/(4*(-a)^(7/8)*b^(1/8)) - (2^(1/2)*atan((2^(1/2)*b^(1/8)*x *(1/2 - 1i/2))/(-a)^(1/8))*(1/8 + 1i/8))/((-a)^(7/8)*b^(1/8)) - (2^(1/2)*a tan((2^(1/2)*b^(1/8)*x*(1/2 + 1i/2))/(-a)^(1/8))*(1/8 - 1i/8))/((-a)^(7/8) *b^(1/8))
Time = 0.22 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.53 \[ \int \frac {1}{a+b x^8} \, dx=\frac {-2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}-2 b^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right )+2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}+2 b^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right )-2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}-2 b^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right )+2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}+2 b^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right )-\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, x +a^{\frac {1}{4}}+b^{\frac {1}{4}} x^{2}\right )+\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, x +a^{\frac {1}{4}}+b^{\frac {1}{4}} x^{2}\right )-\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, x +a^{\frac {1}{4}}+b^{\frac {1}{4}} x^{2}\right )+\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, x +a^{\frac {1}{4}}+b^{\frac {1}{4}} x^{2}\right )}{16 b^{\frac {1}{8}} a^{\frac {7}{8}}} \] Input:
int(1/(b*x^8+a),x)
Output:
(b**(7/8)*a**(1/8)*( - 2*sqrt(sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*b**(1/4)*x)/(b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) + 2*s qrt(sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*b**(1/4) *x)/(b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) - 2*sqrt( - sqrt(2) + 2)*atan(( b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) - 2*b**(1/4)*x)/(b**(1/8)*a**(1/8)*sqr t( - sqrt(2) + 2))) + 2*sqrt( - sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt( sqrt(2) + 2) + 2*b**(1/4)*x)/(b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2))) - s qrt( - sqrt(2) + 2)*log( - b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*x + a**( 1/4) + b**(1/4)*x**2) + sqrt( - sqrt(2) + 2)*log(b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*x + a**(1/4) + b**(1/4)*x**2) - sqrt(sqrt(2) + 2)*log( - b** (1/8)*a**(1/8)*sqrt(sqrt(2) + 2)*x + a**(1/4) + b**(1/4)*x**2) + sqrt(sqrt (2) + 2)*log(b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)*x + a**(1/4) + b**(1/4)*x **2)))/(16*a*b)