Integrand size = 13, antiderivative size = 272 \[ \int \frac {x^8}{a+b x^8} \, dx=\frac {x}{b}-\frac {\sqrt [8]{-a} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}} \]
x/b-1/4*(-a)^(1/8)*arctan(b^(1/8)*x/(-a)^(1/8))/b^(9/8)-1/4*(-a)^(1/8)*arc tanh(b^(1/8)*x/(-a)^(1/8))/b^(9/8)-1/8*(-a)^(1/8)*arctan(-1+b^(1/8)*x*2^(1 /2)/(-a)^(1/8))/b^(9/8)*2^(1/2)-1/8*(-a)^(1/8)*arctan(1+b^(1/8)*x*2^(1/2)/ (-a)^(1/8))/b^(9/8)*2^(1/2)+1/16*(-a)^(1/8)*ln((-a)^(1/4)+b^(1/4)*x^2-(-a) ^(1/8)*b^(1/8)*x*2^(1/2))/b^(9/8)*2^(1/2)-1/16*(-a)^(1/8)*ln((-a)^(1/4)+b^ (1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/b^(9/8)*2^(1/2)
Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int \frac {x^8}{a+b x^8} \, dx=\frac {8 \sqrt [8]{b} x-2 \sqrt [8]{a} \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-2 \sqrt [8]{a} \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+\sqrt [8]{a} \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+2 \sqrt [8]{a} \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )-2 \sqrt [8]{a} \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 b^{9/8}} \]
(8*b^(1/8)*x - 2*a^(1/8)*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]] *Cos[Pi/8] - 2*a^(1/8)*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*C os[Pi/8] + a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8) *x*Cos[Pi/8]] - a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^ (1/8)*x*Cos[Pi/8]] + 2*a^(1/8)*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^ (1/8)]*Sin[Pi/8] - 2*a^(1/8)*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1 /8)]*Sin[Pi/8] + a^(1/8)*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*S in[Pi/8]]*Sin[Pi/8] - a^(1/8)*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8 )*x*Sin[Pi/8]]*Sin[Pi/8])/(8*b^(9/8))
Time = 0.58 (sec) , antiderivative size = 322, 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 = {843, 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.
| \(\displaystyle \int \frac {x^8}{a+b x^8} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x}{b}-\frac {a \int \frac {1}{b x^8+a}dx}{b}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4}dx}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{b} x^2+\sqrt [4]{-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{-a}}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {-a}}dx}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{b} x}{x^2-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{-a}}{x^2+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}}dx}{2 \sqrt [8]{-a} \sqrt [4]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (-\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}\) |
x/b - (a*(-1/2*(ArcTan[(b^(1/8)*x)/(-a)^(1/8)]/(2*(-a)^(3/8)*b^(1/8)) + Ar cTanh[(b^(1/8)*x)/(-a)^(1/8)]/(2*(-a)^(3/8)*b^(1/8)))/Sqrt[-a] - ((-(ArcTa n[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8))) + ArcT an[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])))/b
3.15.61.3.1 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^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && 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 = 5.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.12
| method | result | size |
| default | \(\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8 b^{2}}\) | \(34\) |
| risch | \(\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8 b^{2}}\) | \(34\) |
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.85 \[ \int \frac {x^8}{a+b x^8} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - \left (i - 1\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) + \left (i - 1\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - \left (i + 1\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) + 2 \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) + 2 i \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (i \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - 2 i \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-i \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - 2 \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - 16 \, x}{16 \, b} \]
-1/16*((I + 1)*sqrt(2)*b*(-a/b^9)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*b*(-a/b^ 9)^(1/8) + x) - (I - 1)*sqrt(2)*b*(-a/b^9)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2 )*b*(-a/b^9)^(1/8) + x) + (I - 1)*sqrt(2)*b*(-a/b^9)^(1/8)*log((1/2*I - 1/ 2)*sqrt(2)*b*(-a/b^9)^(1/8) + x) - (I + 1)*sqrt(2)*b*(-a/b^9)^(1/8)*log(-( 1/2*I + 1/2)*sqrt(2)*b*(-a/b^9)^(1/8) + x) + 2*b*(-a/b^9)^(1/8)*log(b*(-a/ b^9)^(1/8) + x) + 2*I*b*(-a/b^9)^(1/8)*log(I*b*(-a/b^9)^(1/8) + x) - 2*I*b *(-a/b^9)^(1/8)*log(-I*b*(-a/b^9)^(1/8) + x) - 2*b*(-a/b^9)^(1/8)*log(-b*( -a/b^9)^(1/8) + x) - 16*x)/b
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.08 \[ \int \frac {x^8}{a+b x^8} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} b^{9} + a, \left ( t \mapsto t \log {\left (- 8 t b + x \right )} \right )\right )} + \frac {x}{b} \]
\[ \int \frac {x^8}{a+b x^8} \, dx=\int { \frac {x^{8}}{b x^{8} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (187) = 374\).
Time = 0.31 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.62 \[ \int \frac {x^8}{a+b x^8} \, dx=\frac {x}{b} - \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 \, b \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 \, b \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 \, b \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 \, b \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 \, b \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 \, b \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 \, b \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 \, b \sqrt {2 \, \sqrt {2} + 4}} \]
x/b - 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)))/(b*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/b)^(1/8)*arct an((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8))) /(b*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)))/(b*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)))/(b*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))/(b*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))/( b*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))/(b*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))/(b*sqrt(2*sqrt(2) + 4) )
Time = 5.89 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.42 \[ \int \frac {x^8}{a+b x^8} \, dx=\frac {x}{b}-\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {b^{1/8}\,x}{{\left (-a\right )}^{1/8}}\right )}{4\,b^{9/8}}+\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,b^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{b^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{b^{9/8}} \]
x/b - ((-a)^(1/8)*atan((b^(1/8)*x)/(-a)^(1/8)))/(4*b^(9/8)) + ((-a)^(1/8)* atan((b^(1/8)*x*1i)/(-a)^(1/8))*1i)/(4*b^(9/8)) - (2^(1/2)*(-a)^(1/8)*atan ((2^(1/2)*b^(1/8)*x*(1/2 - 1i/2))/(-a)^(1/8))*(1/8 + 1i/8))/b^(9/8) - (2^( 1/2)*(-a)^(1/8)*atan((2^(1/2)*b^(1/8)*x*(1/2 + 1i/2))/(-a)^(1/8))*(1/8 - 1 i/8))/b^(9/8)