Integrand size = 13, antiderivative size = 277 \[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=-\frac {1}{5 a x^5}+\frac {b^{5/8} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}+\frac {b^{5/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {b^{5/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}} \]
-1/5/a/x^5+1/4*b^(5/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(13/8)-1/4*b^(5/8 )*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)^(13/8)-1/8*b^(5/8)*arctan(-1+b^(1/8)* x*2^(1/2)/(-a)^(1/8))/(-a)^(13/8)*2^(1/2)-1/8*b^(5/8)*arctan(1+b^(1/8)*x*2 ^(1/2)/(-a)^(1/8))/(-a)^(13/8)*2^(1/2)-1/16*b^(5/8)*ln((-a)^(1/4)+b^(1/4)* x^2-(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(13/8)*2^(1/2)+1/16*b^(5/8)*ln((-a) ^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(13/8)*2^(1/2)
Time = 0.10 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=\frac {-8 a^{5/8}+10 b^{5/8} x^5 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \cos \left (\frac {\pi }{8}\right )-10 b^{5/8} x^5 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \cos \left (\frac {\pi }{8}\right )-5 b^{5/8} x^5 \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+5 b^{5/8} x^5 \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+10 b^{5/8} x^5 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+10 b^{5/8} x^5 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+5 b^{5/8} x^5 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-5 b^{5/8} x^5 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{40 a^{13/8} x^5} \]
(-8*a^(5/8) + 10*b^(5/8)*x^5*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1 /8)]*Cos[Pi/8] - 10*b^(5/8)*x^5*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a ^(1/8)]*Cos[Pi/8] - 5*b^(5/8)*x^5*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2* a^(1/8)*b^(1/8)*x*Sin[Pi/8]] + 5*b^(5/8)*x^5*Cos[Pi/8]*Log[a^(1/4) + b^(1/ 4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] + 10*b^(5/8)*x^5*ArcTan[(b^(1/8)*x *Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8] + 10*b^(5/8)*x^5*ArcTan[(b^(1/8 )*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[Pi/8] + 5*b^(5/8)*x^5*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]]*Sin[Pi/8] - 5*b^(5/8)*x^5*L og[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]]*Sin[Pi/8])/(40*a ^(13/8)*x^5)
Time = 0.54 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {847, 829, 826, 827, 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.
| \(\displaystyle \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {b \int \frac {x^2}{b x^8+a}dx}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 829 |
| \(\displaystyle -\frac {b \left (-\frac {\int \frac {x^2}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\int \frac {x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {b \left (-\frac {\int \frac {x^2}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{b}}-\frac {\int \frac {1}{\sqrt [4]{b} x^2+\sqrt [4]{-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [8]{-a} \sqrt [4]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {b \left (-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} b^{3/8}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {-a}}\right )}{a}-\frac {1}{5 a x^5}\) |
-1/5*1/(a*x^5) - (b*(-1/2*(-1/2*ArcTan[(b^(1/8)*x)/(-a)^(1/8)]/((-a)^(1/8) *b^(3/8)) + ArcTanh[(b^(1/8)*x)/(-a)^(1/8)]/(2*(-a)^(1/8)*b^(3/8)))/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*b^(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*b^(1/ 4)))/(2*Sqrt[-a])))/a
3.15.68.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[x^m/(r + s*x^ (n/2)), x], x] + Simp[r/(2*a) Int[x^m/(r - s*x^(n/2)), x], x]] /; FreeQ[{ a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && !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 = 3.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(-\frac {1}{5 a \,x^{5}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{8 a}\) | \(36\) |
| risch | \(-\frac {1}{5 a \,x^{5}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{8}+b^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (-9 \textit {\_R}^{8} a^{13}-8 b^{5}\right ) x +a^{5} b^{3} \textit {\_R}^{3}\right )\right )}{8}\) | \(57\) |
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=-\frac {-\left (5 i - 5\right ) \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) + \left (5 i + 5\right ) \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) - \left (5 i + 5\right ) \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) + \left (5 i - 5\right ) \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) - 10 \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) + 10 i \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (i \, a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) - 10 i \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (-i \, a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) + 10 \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (-a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) + 16}{80 \, a x^{5}} \]
-1/80*(-(5*I - 5)*sqrt(2)*a*x^5*(-b^5/a^13)^(1/8)*log((1/2*I + 1/2)*sqrt(2 )*a^5*(-b^5/a^13)^(3/8) + b^2*x) + (5*I + 5)*sqrt(2)*a*x^5*(-b^5/a^13)^(1/ 8)*log(-(1/2*I - 1/2)*sqrt(2)*a^5*(-b^5/a^13)^(3/8) + b^2*x) - (5*I + 5)*s qrt(2)*a*x^5*(-b^5/a^13)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^5*(-b^5/a^13)^( 3/8) + b^2*x) + (5*I - 5)*sqrt(2)*a*x^5*(-b^5/a^13)^(1/8)*log(-(1/2*I + 1/ 2)*sqrt(2)*a^5*(-b^5/a^13)^(3/8) + b^2*x) - 10*a*x^5*(-b^5/a^13)^(1/8)*log (a^5*(-b^5/a^13)^(3/8) + b^2*x) + 10*I*a*x^5*(-b^5/a^13)^(1/8)*log(I*a^5*( -b^5/a^13)^(3/8) + b^2*x) - 10*I*a*x^5*(-b^5/a^13)^(1/8)*log(-I*a^5*(-b^5/ a^13)^(3/8) + b^2*x) + 10*a*x^5*(-b^5/a^13)^(1/8)*log(-a^5*(-b^5/a^13)^(3/ 8) + b^2*x) + 16)/(a*x^5)
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{13} + b^{5}, \left ( t \mapsto t \log {\left (\frac {512 t^{3} a^{5}}{b^{2}} + x \right )} \right )\right )} - \frac {1}{5 a x^{5}} \]
RootSum(16777216*_t**8*a**13 + b**5, Lambda(_t, _t*log(512*_t**3*a**5/b**2 + x))) - 1/(5*a*x**5)
\[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} x^{6}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (190) = 380\).
Time = 0.38 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.64 \[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=\frac {b \left (\frac {a}{b}\right )^{\frac {3}{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 {3}{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 {3}{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 {3}{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 {3}{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 {3}{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 {3}{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 {3}{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}{5 \, a x^{5}} \]
1/4*b*(a/b)^(3/8)*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)^(3/8)*arcta n((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)^(3/8)*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)^(3/8)*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)) - 1/8*b*(a/b)^(3/8 )*log(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a^2*sqrt(2*sqr t(2) + 4)) + 1/8*b*(a/b)^(3/8)*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)^(3/8)*log(x^2 + x*sq rt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1 /8*b*(a/b)^(3/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4)) /(a^2*sqrt(-2*sqrt(2) + 4)) - 1/5/(a*x^5)
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx=-\frac {1}{5\,a\,x^5}-\frac {{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{13/8}}-\frac {{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{13/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{13/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{13/8}} \]
(2^(1/2)*(-b)^(5/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 - 1i/2))/a^(1/8))*(1/8 - 1i/8))/a^(13/8) - ((-b)^(5/8)*atan(((-b)^(1/8)*x)/a^(1/8)))/(4*a^(13/8) ) - ((-b)^(5/8)*atan(((-b)^(1/8)*x*1i)/a^(1/8))*1i)/(4*a^(13/8)) - 1/(5*a* x^5) + (2^(1/2)*(-b)^(5/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 + 1i/2))/a^(1/8 ))*(1/8 + 1i/8))/a^(13/8)