Integrand size = 9, antiderivative size = 557 \[ \int \frac {1}{b+a x^8} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {-\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}-\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 \sqrt [8]{a} b^{7/8}}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {-\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}+\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 \sqrt [8]{a} b^{7/8}}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}-\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 \sqrt [8]{a} b^{7/8}}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}+\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 \sqrt [8]{a} b^{7/8}}-\frac {\sqrt {2-\sqrt {2}} \log \left (\sqrt [4]{b}-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 \sqrt [8]{a} b^{7/8}}+\frac {\sqrt {2-\sqrt {2}} \log \left (\sqrt [4]{b}+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 \sqrt [8]{a} b^{7/8}}-\frac {\sqrt {2+\sqrt {2}} \log \left (\sqrt [4]{b}-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 \sqrt [8]{a} b^{7/8}}+\frac {\sqrt {2+\sqrt {2}} \log \left (\sqrt [4]{b}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 \sqrt [8]{a} b^{7/8}} \] Output:
-1/8*(2+2^(1/2))^(1/2)*arctan((-b^(1/8)+b^(1/8)*2^(1/2)-(4-2*2^(1/2))^(1/2 )*a^(1/8)*x)/b^(1/8))/a^(1/8)/b^(7/8)+1/8*(2+2^(1/2))^(1/2)*arctan((-b^(1/ 8)+b^(1/8)*2^(1/2)+(4-2*2^(1/2))^(1/2)*a^(1/8)*x)/b^(1/8))/a^(1/8)/b^(7/8) -1/8*(2-2^(1/2))^(1/2)*arctan((b^(1/8)+b^(1/8)*2^(1/2)-(4+2*2^(1/2))^(1/2) *a^(1/8)*x)/b^(1/8))/a^(1/8)/b^(7/8)+1/8*(2-2^(1/2))^(1/2)*arctan((b^(1/8) +b^(1/8)*2^(1/2)+(4+2*2^(1/2))^(1/2)*a^(1/8)*x)/b^(1/8))/a^(1/8)/b^(7/8)-1 /16*(2-2^(1/2))^(1/2)*ln(b^(1/4)-(2-2^(1/2))^(1/2)*a^(1/8)*b^(1/8)*x+a^(1/ 4)*x^2)/a^(1/8)/b^(7/8)+1/16*(2-2^(1/2))^(1/2)*ln(b^(1/4)+(2-2^(1/2))^(1/2 )*a^(1/8)*b^(1/8)*x+a^(1/4)*x^2)/a^(1/8)/b^(7/8)-1/16*(2+2^(1/2))^(1/2)*ln (b^(1/4)-(2+2^(1/2))^(1/2)*a^(1/8)*b^(1/8)*x+a^(1/4)*x^2)/a^(1/8)/b^(7/8)+ 1/16*(2+2^(1/2))^(1/2)*ln(b^(1/4)+(2+2^(1/2))^(1/2)*a^(1/8)*b^(1/8)*x+a^(1 /4)*x^2)/a^(1/8)/b^(7/8)
Time = 0.17 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.58 \[ \int \frac {1}{b+a x^8} \, dx=\frac {2 \arctan \left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+2 \arctan \left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-2 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right ) \sin \left (\frac {\pi }{8}\right )+2 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right ) \sin \left (\frac {\pi }{8}\right )-\log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 \sqrt [8]{a} b^{7/8}} \] Input:
Integrate[(b + a*x^8)^(-1),x]
Output:
(2*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) - Tan[Pi/8]]*Cos[Pi/8] + 2*ArcTan[ (a^(1/8)*x*Sec[Pi/8])/b^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - Cos[Pi/8]*Log[b^(1/ 4) + a^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - 2*ArcTan[Cot[Pi/8] - (a^( 1/8)*x*Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] + 2*ArcTan[Cot[Pi/8] + (a^(1/8)*x*Csc [Pi/8])/b^(1/8)]*Sin[Pi/8] - Log[b^(1/4) + a^(1/4)*x^2 - 2*a^(1/8)*b^(1/8) *x*Sin[Pi/8]]*Sin[Pi/8] + Log[b^(1/4) + a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x* Sin[Pi/8]]*Sin[Pi/8])/(8*a^(1/8)*b^(7/8))
Time = 0.99 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.54, 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.
| \(\displaystyle \int \frac {1}{a x^8+b} \, dx\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {b}-\sqrt {-a} x^4}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {b}-\sqrt {-a} x^4}dx}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\sqrt [4]{b}-\sqrt [4]{-a} x^2}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{\sqrt [4]{-a} x^2+\sqrt [4]{b}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\sqrt [4]{b}-\sqrt [4]{-a} x^2}dx}{2 \sqrt [4]{b}}+\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} x+\frac {\sqrt [8]{b}}{\sqrt [8]{-a}}\right )}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} x+\frac {\sqrt [8]{b}}{\sqrt [8]{-a}}\right )}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} x+\frac {\sqrt [8]{b}}{\sqrt [8]{-a}}}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}\) |
Input:
Int[(b + a*x^8)^(-1),x]
Output:
(ArcTan[((-a)^(1/8)*x)/b^(1/8)]/(2*(-a)^(1/8)*b^(3/8)) + ArcTanh[((-a)^(1/ 8)*x)/b^(1/8)]/(2*(-a)^(1/8)*b^(3/8)))/(2*Sqrt[b]) + ((-(ArcTan[1 - (Sqrt[ 2]*(-a)^(1/8)*x)/b^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8))) + ArcTan[1 + (Sqrt [2]*(-a)^(1/8)*x)/b^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8)))/(2*b^(1/4)) + (-1 /2*Log[b^(1/4) - Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + (-a)^(1/4)*x^2]/(Sqrt[2]*( -a)^(1/8)*b^(1/8)) + Log[b^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + (-a)^(1/ 4)*x^2]/(2*Sqrt[2]*(-a)^(1/8)*b^(1/8)))/(2*b^(1/4)))/(2*Sqrt[b])
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 = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.05
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 a}\) | \(27\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 a}\) | \(27\) |
Input:
int(1/(a*x^8+b),x,method=_RETURNVERBOSE)
Output:
1/8/a*sum(1/_R^7*ln(x-_R),_R=RootOf(_Z^8*a+b))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.45 \[ \int \frac {1}{b+a x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} \, \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} i \, \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (i \, b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} i \, \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (-i \, b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} \, \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} \log \left (-b \left (-\frac {1}{a b^{7}}\right )^{\frac {1}{8}} + x\right ) \] Input:
integrate(1/(a*x^8+b),x, algorithm="fricas")
Output:
(1/16*I + 1/16)*sqrt(2)*(-1/(a*b^7))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*b*(-1 /(a*b^7))^(1/8) + x) - (1/16*I - 1/16)*sqrt(2)*(-1/(a*b^7))^(1/8)*log(-(1/ 2*I - 1/2)*sqrt(2)*b*(-1/(a*b^7))^(1/8) + x) + (1/16*I - 1/16)*sqrt(2)*(-1 /(a*b^7))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*b*(-1/(a*b^7))^(1/8) + x) - (1/1 6*I + 1/16)*sqrt(2)*(-1/(a*b^7))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*b*(-1/(a *b^7))^(1/8) + x) + 1/8*(-1/(a*b^7))^(1/8)*log(b*(-1/(a*b^7))^(1/8) + x) + 1/8*I*(-1/(a*b^7))^(1/8)*log(I*b*(-1/(a*b^7))^(1/8) + x) - 1/8*I*(-1/(a*b ^7))^(1/8)*log(-I*b*(-1/(a*b^7))^(1/8) + x) - 1/8*(-1/(a*b^7))^(1/8)*log(- b*(-1/(a*b^7))^(1/8) + x)
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.04 \[ \int \frac {1}{b+a x^8} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a b^{7} + 1, \left ( t \mapsto t \log {\left (8 t b + x \right )} \right )\right )} \] Input:
integrate(1/(a*x**8+b),x)
Output:
RootSum(16777216*_t**8*a*b**7 + 1, Lambda(_t, _t*log(8*_t*b + x)))
\[ \int \frac {1}{b+a x^8} \, dx=\int { \frac {1}{a x^{8} + b} \,d x } \] Input:
integrate(1/(a*x^8+b),x, algorithm="maxima")
Output:
integrate(1/(a*x^8 + b), x)
Time = 0.15 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.78 \[ \int \frac {1}{b+a x^8} \, dx=\frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {2 \, \sqrt {2} + 4}} \] Input:
integrate(1/(a*x^8+b),x, algorithm="giac")
Output:
1/4*(b/a)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(sqrt(2 ) + 2)*(b/a)^(1/8)))/(b*sqrt(-2*sqrt(2) + 4)) + 1/4*(b/a)^(1/8)*arctan((2* x - sqrt(-sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(sqrt(2) + 2)*(b/a)^(1/8)))/(b*sq rt(-2*sqrt(2) + 4)) + 1/4*(b/a)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(b/a )^(1/8))/(sqrt(-sqrt(2) + 2)*(b/a)^(1/8)))/(b*sqrt(2*sqrt(2) + 4)) + 1/4*( b/a)^(1/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(-sqrt(2) + 2 )*(b/a)^(1/8)))/(b*sqrt(2*sqrt(2) + 4)) + 1/8*(b/a)^(1/8)*log(x^2 + x*sqrt (sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/(b*sqrt(-2*sqrt(2) + 4)) - 1/8*(b /a)^(1/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/(b*sqrt (-2*sqrt(2) + 4)) + 1/8*(b/a)^(1/8)*log(x^2 + x*sqrt(-sqrt(2) + 2)*(b/a)^( 1/8) + (b/a)^(1/4))/(b*sqrt(2*sqrt(2) + 4)) - 1/8*(b/a)^(1/8)*log(x^2 - x* sqrt(-sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/(b*sqrt(2*sqrt(2) + 4))
Time = 9.91 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.20 \[ \int \frac {1}{b+a x^8} \, dx=\frac {\mathrm {atan}\left (\frac {{\left (-a\right )}^{1/8}\,x}{b^{1/8}}\right )}{4\,{\left (-a\right )}^{1/8}\,b^{7/8}}-\frac {\mathrm {atan}\left (\frac {{\left (-a\right )}^{1/8}\,x\,1{}\mathrm {i}}{b^{1/8}}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{1/8}\,b^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{b^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,b^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{b^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,b^{7/8}} \] Input:
int(1/(b + a*x^8),x)
Output:
atan(((-a)^(1/8)*x)/b^(1/8))/(4*(-a)^(1/8)*b^(7/8)) - (atan(((-a)^(1/8)*x* 1i)/b^(1/8))*1i)/(4*(-a)^(1/8)*b^(7/8)) + (2^(1/2)*atan((2^(1/2)*(-a)^(1/8 )*x*(1/2 - 1i/2))/b^(1/8))*(1/8 + 1i/8))/((-a)^(1/8)*b^(7/8)) + (2^(1/2)*a tan((2^(1/2)*(-a)^(1/8)*x*(1/2 + 1i/2))/b^(1/8))*(1/8 - 1i/8))/((-a)^(1/8) *b^(7/8))
Time = 0.25 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.58 \[ \int \frac {1}{b+a x^8} \, dx=\frac {-2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}-2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right )+2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}+2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right )-2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}-2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right )+2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}+2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right )-\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right )+\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right )-\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right )+\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right )}{16 b^{\frac {7}{8}} a^{\frac {1}{8}}} \] Input:
int(1/(a*x^8+b),x)
Output:
(b**(1/8)*a**(7/8)*( - 2*sqrt(sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*a**(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*a**(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*a**(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*a**(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)*x**2 + b**(1/4)) + sqrt( - sqrt(2) + 2)*log(b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*x + a**(1/4)*x**2 + b**(1/4)) - sqrt(sqrt(2) + 2)*log( - b** (1/8)*a**(1/8)*sqrt(sqrt(2) + 2)*x + a**(1/4)*x**2 + b**(1/4)) + sqrt(sqrt (2) + 2)*log(b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)*x + a**(1/4)*x**2 + b**(1 /4))))/(16*a*b)