Integrand size = 13, antiderivative size = 277 \[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=-\frac {1}{7 a x^7}-\frac {b^{7/8} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}} \]
-1/7/a/x^7-1/4*b^(7/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(15/8)-1/4*b^(7/8 )*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)^(15/8)-1/8*b^(7/8)*arctan(-1+b^(1/8)* x*2^(1/2)/(-a)^(1/8))/(-a)^(15/8)*2^(1/2)-1/8*b^(7/8)*arctan(1+b^(1/8)*x*2 ^(1/2)/(-a)^(1/8))/(-a)^(15/8)*2^(1/2)+1/16*b^(7/8)*ln((-a)^(1/4)+b^(1/4)* x^2-(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(15/8)*2^(1/2)-1/16*b^(7/8)*ln((-a) ^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(15/8)*2^(1/2)
Time = 0.14 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=-\frac {8 a^{7/8}+14 b^{7/8} x^7 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+14 b^{7/8} x^7 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-7 b^{7/8} x^7 \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+7 b^{7/8} x^7 \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-14 b^{7/8} x^7 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+14 b^{7/8} x^7 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )-7 b^{7/8} x^7 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+7 b^{7/8} x^7 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{56 a^{15/8} x^7} \]
-1/56*(8*a^(7/8) + 14*b^(7/8)*x^7*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - T an[Pi/8]]*Cos[Pi/8] + 14*b^(7/8)*x^7*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - 7*b^(7/8)*x^7*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 7*b^(7/8)*x^7*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - 14*b^(7/8)*x^7*ArcTan[Cot[P i/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + 14*b^(7/8)*x^7*ArcTan[Co t[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - 7*b^(7/8)*x^7*Log[a^( 1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] + 7*b^(7/8)* x^7*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/ (a^(15/8)*x^7)
Time = 0.53 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.18, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {847, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {b \int \frac {1}{b x^8+a}dx}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle -\frac {b \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {b \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{b} x^2+\sqrt [4]{-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [8]{-a} \sqrt [4]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{7 a x^7}\) |
-1/7*1/(a*x^7) - (b*(-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 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/ 8))) + ArcTan[1 + (Sqrt[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])))/a
3.15.69.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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 = 5.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(-\frac {1}{7 a \,x^{7}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 a}\) | \(36\) |
| risch | \(-\frac {1}{7 a \,x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{15} \textit {\_Z}^{8}+b^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (-9 \textit {\_R}^{8} a^{15}-8 b^{7}\right ) x -a^{2} b^{6} \textit {\_R} \right )\right )}{8}\) | \(56\) |
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=-\frac {\left (7 i + 7\right ) \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) - \left (7 i - 7\right ) \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) + \left (7 i - 7\right ) \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) - \left (7 i + 7\right ) \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) + 14 \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) + 14 i \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (i \, a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) - 14 i \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (-i \, a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) - 14 \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (-a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) + 16}{112 \, a x^{7}} \]
-1/112*((7*I + 7)*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8)*log((1/2*I + 1/2)*sqrt(2 )*a^2*(-b^7/a^15)^(1/8) + b*x) - (7*I - 7)*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8) *log(-(1/2*I - 1/2)*sqrt(2)*a^2*(-b^7/a^15)^(1/8) + b*x) + (7*I - 7)*sqrt( 2)*a*x^7*(-b^7/a^15)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^2*(-b^7/a^15)^(1/8) + b*x) - (7*I + 7)*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8)*log(-(1/2*I + 1/2)*sqr t(2)*a^2*(-b^7/a^15)^(1/8) + b*x) + 14*a*x^7*(-b^7/a^15)^(1/8)*log(a^2*(-b ^7/a^15)^(1/8) + b*x) + 14*I*a*x^7*(-b^7/a^15)^(1/8)*log(I*a^2*(-b^7/a^15) ^(1/8) + b*x) - 14*I*a*x^7*(-b^7/a^15)^(1/8)*log(-I*a^2*(-b^7/a^15)^(1/8) + b*x) - 14*a*x^7*(-b^7/a^15)^(1/8)*log(-a^2*(-b^7/a^15)^(1/8) + b*x) + 16 )/(a*x^7)
Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{15} + b^{7}, \left ( t \mapsto t \log {\left (- \frac {8 t a^{2}}{b} + x \right )} \right )\right )} - \frac {1}{7 a x^{7}} \]
\[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} x^{8}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (190) = 380\).
Time = 0.28 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.64 \[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=-\frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {1}{7 \, a x^{7}} \]
-1/4*b*(a/b)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqr t(2) + 2)*(a/b)^(1/8)))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/4*b*(a/b)^(1/8)*arc tan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)) )/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/4*b*(a/b)^(1/8)*arctan((2*x + sqrt(sqrt(2 ) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/4*b*(a/b)^(1/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sq rt(-sqrt(2) + 2)*(a/b)^(1/8)))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/8*b*(a/b)^(1/ 8)*log(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a^2*sqrt(-2*s qrt(2) + 4)) + 1/8*b*(a/b)^(1/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/8*b*(a/b)^(1/8)*log(x^2 + x *sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a^2*sqrt(2*sqrt(2) + 4)) + 1/8*b*(a/b)^(1/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4 ))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/7/(a*x^7)
Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx=\frac {{\left (-b\right )}^{7/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{15/8}}-\frac {1}{7\,a\,x^7}-\frac {{\left (-b\right )}^{7/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{15/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{7/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{15/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{7/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{15/8}} \]
((-b)^(7/8)*atan(((-b)^(1/8)*x)/a^(1/8)))/(4*a^(15/8)) - 1/(7*a*x^7) - ((- b)^(7/8)*atan(((-b)^(1/8)*x*1i)/a^(1/8))*1i)/(4*a^(15/8)) + (2^(1/2)*(-b)^ (7/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 - 1i/2))/a^(1/8))*(1/8 + 1i/8))/a^(1 5/8) + (2^(1/2)*(-b)^(7/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 + 1i/2))/a^(1/8 ))*(1/8 - 1i/8))/a^(15/8)