Integrand size = 9, antiderivative size = 335 \[ \int \frac {1}{a+b x^7} \, dx=\frac {2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {2 \arctan \left (\cot \left (\frac {\pi }{7}\right )-\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac {\pi }{7}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
2/7*arctan(b^(1/7)*x*sec(1/14*Pi)/a^(1/7)-tan(1/14*Pi))*cos(1/14*Pi)/a^(6/ 7)/b^(1/7)+2/7*arctan(b^(1/7)*x*sec(3/14*Pi)/a^(1/7)+tan(3/14*Pi))*cos(3/1 4*Pi)/a^(6/7)/b^(1/7)+1/7*ln(a^(1/7)+b^(1/7)*x)/a^(6/7)/b^(1/7)-1/7*cos(1/ 7*Pi)*ln(a^(2/7)+b^(2/7)*x^2-2*a^(1/7)*b^(1/7)*x*cos(1/7*Pi))/a^(6/7)/b^(1 /7)-1/7*ln(a^(2/7)+b^(2/7)*x^2-2*a^(1/7)*b^(1/7)*x*sin(1/14*Pi))*sin(1/14* Pi)/a^(6/7)/b^(1/7)+2/7*arctan(-cot(1/7*Pi)+b^(1/7)*x*csc(1/7*Pi)/a^(1/7)) *sin(1/7*Pi)/a^(6/7)/b^(1/7)+1/7*ln(a^(2/7)+b^(2/7)*x^2+2*a^(1/7)*b^(1/7)* x*sin(3/14*Pi))*sin(3/14*Pi)/a^(6/7)/b^(1/7)
Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a+b x^7} \, dx=\frac {2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )+2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )-\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )-\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-2 \arctan \left (\cot \left (\frac {\pi }{7}\right )-\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac {\pi }{7}\right )+\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
(2*ArcTan[(b^(1/7)*x*Sec[Pi/14])/a^(1/7) - Tan[Pi/14]]*Cos[Pi/14] + 2*ArcT an[(b^(1/7)*x*Sec[(3*Pi)/14])/a^(1/7) + Tan[(3*Pi)/14]]*Cos[(3*Pi)/14] + L og[a^(1/7) + b^(1/7)*x] - Cos[Pi/7]*Log[a^(2/7) + b^(2/7)*x^2 - 2*a^(1/7)* b^(1/7)*x*Cos[Pi/7]] - Log[a^(2/7) + b^(2/7)*x^2 - 2*a^(1/7)*b^(1/7)*x*Sin [Pi/14]]*Sin[Pi/14] - 2*ArcTan[Cot[Pi/7] - (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]* Sin[Pi/7] + Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Sin[(3*Pi)/14] ]*Sin[(3*Pi)/14])/(7*a^(6/7)*b^(1/7))
Time = 0.78 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {751, 16, 1142, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{a+b x^7} \, dx\) |
| \(\Big \downarrow \) 751 |
| \(\displaystyle \frac {2 \int \frac {\sqrt [7]{a}-\sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx}{7 a^{6/7}}+\frac {2 \int \frac {\sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+\sqrt [7]{a}}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx}{7 a^{6/7}}+\frac {2 \int \frac {\sqrt [7]{a}-\sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx}{7 a^{6/7}}+\frac {\int \frac {1}{\sqrt [7]{b} x+\sqrt [7]{a}}dx}{7 a^{6/7}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \int \frac {\sqrt [7]{a}-\sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx}{7 a^{6/7}}+\frac {2 \int \frac {\sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+\sqrt [7]{a}}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx}{7 a^{6/7}}+\frac {2 \int \frac {\sqrt [7]{a}-\sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx}{7 a^{6/7}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2 \left (\sqrt [7]{a} \cos ^2\left (\frac {\pi }{14}\right ) \int \frac {1}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx-\frac {\sin \left (\frac {\pi }{14}\right ) \int \frac {2 \sqrt [7]{b} \left (\sqrt [7]{b} x-\sqrt [7]{a} \sin \left (\frac {\pi }{14}\right )\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx}{2 \sqrt [7]{b}}\right )}{7 a^{6/7}}+\frac {2 \left (\frac {\sin \left (\frac {3 \pi }{14}\right ) \int \frac {2 \sqrt [7]{b} \left (\sqrt [7]{b} x+\sqrt [7]{a} \sin \left (\frac {3 \pi }{14}\right )\right )}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx}{2 \sqrt [7]{b}}+\sqrt [7]{a} \cos ^2\left (\frac {3 \pi }{14}\right ) \int \frac {1}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx\right )}{7 a^{6/7}}+\frac {2 \left (\sqrt [7]{a} \sin ^2\left (\frac {\pi }{7}\right ) \int \frac {1}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx-\frac {\cos \left (\frac {\pi }{7}\right ) \int \frac {2 \sqrt [7]{b} \left (\sqrt [7]{b} x-\sqrt [7]{a} \cos \left (\frac {\pi }{7}\right )\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx}{2 \sqrt [7]{b}}\right )}{7 a^{6/7}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\sqrt [7]{a} \cos ^2\left (\frac {\pi }{14}\right ) \int \frac {1}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx-\sin \left (\frac {\pi }{14}\right ) \int \frac {\sqrt [7]{b} x-\sqrt [7]{a} \sin \left (\frac {\pi }{14}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx\right )}{7 a^{6/7}}+\frac {2 \left (\sin \left (\frac {3 \pi }{14}\right ) \int \frac {\sqrt [7]{b} x+\sqrt [7]{a} \sin \left (\frac {3 \pi }{14}\right )}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx+\sqrt [7]{a} \cos ^2\left (\frac {3 \pi }{14}\right ) \int \frac {1}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx\right )}{7 a^{6/7}}+\frac {2 \left (\sqrt [7]{a} \sin ^2\left (\frac {\pi }{7}\right ) \int \frac {1}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx-\cos \left (\frac {\pi }{7}\right ) \int \frac {\sqrt [7]{b} x-\sqrt [7]{a} \cos \left (\frac {\pi }{7}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx\right )}{7 a^{6/7}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2 \left (-\sin \left (\frac {\pi }{14}\right ) \int \frac {\sqrt [7]{b} x-\sqrt [7]{a} \sin \left (\frac {\pi }{14}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx-2 \sqrt [7]{a} \cos ^2\left (\frac {\pi }{14}\right ) \int \frac {1}{-\left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right )\right )^2-4 a^{2/7} b^{2/7} \cos ^2\left (\frac {\pi }{14}\right )}d\left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right )\right )\right )}{7 a^{6/7}}+\frac {2 \left (\sin \left (\frac {3 \pi }{14}\right ) \int \frac {\sqrt [7]{b} x+\sqrt [7]{a} \sin \left (\frac {3 \pi }{14}\right )}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx-2 \sqrt [7]{a} \cos ^2\left (\frac {3 \pi }{14}\right ) \int \frac {1}{-\left (2 b^{2/7} x+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right )\right )^2-4 a^{2/7} b^{2/7} \cos ^2\left (\frac {3 \pi }{14}\right )}d\left (2 b^{2/7} x+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right )\right )\right )}{7 a^{6/7}}+\frac {2 \left (-\cos \left (\frac {\pi }{7}\right ) \int \frac {\sqrt [7]{b} x-\sqrt [7]{a} \cos \left (\frac {\pi }{7}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx-2 \sqrt [7]{a} \sin ^2\left (\frac {\pi }{7}\right ) \int \frac {1}{-\left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right )\right )^2-4 a^{2/7} b^{2/7} \sin ^2\left (\frac {\pi }{7}\right )}d\left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right )\right )\right )}{7 a^{6/7}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {\sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {\csc \left (\frac {\pi }{7}\right ) \left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right )\right )}{2 \sqrt [7]{a} \sqrt [7]{b}}\right )}{\sqrt [7]{b}}-\cos \left (\frac {\pi }{7}\right ) \int \frac {\sqrt [7]{b} x-\sqrt [7]{a} \cos \left (\frac {\pi }{7}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right ) x+a^{2/7}}dx\right )}{7 a^{6/7}}+\frac {2 \left (\frac {\cos \left (\frac {\pi }{14}\right ) \arctan \left (\frac {\sec \left (\frac {\pi }{14}\right ) \left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right )\right )}{2 \sqrt [7]{a} \sqrt [7]{b}}\right )}{\sqrt [7]{b}}-\sin \left (\frac {\pi }{14}\right ) \int \frac {\sqrt [7]{b} x-\sqrt [7]{a} \sin \left (\frac {\pi }{14}\right )}{b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right ) x+a^{2/7}}dx\right )}{7 a^{6/7}}+\frac {2 \left (\sin \left (\frac {3 \pi }{14}\right ) \int \frac {\sqrt [7]{b} x+\sqrt [7]{a} \sin \left (\frac {3 \pi }{14}\right )}{b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right ) x+a^{2/7}}dx+\frac {\cos \left (\frac {3 \pi }{14}\right ) \arctan \left (\frac {\sec \left (\frac {3 \pi }{14}\right ) \left (2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right )+2 b^{2/7} x\right )}{2 \sqrt [7]{a} \sqrt [7]{b}}\right )}{\sqrt [7]{b}}\right )}{7 a^{6/7}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {\cos \left (\frac {\pi }{14}\right ) \arctan \left (\frac {\sec \left (\frac {\pi }{14}\right ) \left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right )\right )}{2 \sqrt [7]{a} \sqrt [7]{b}}\right )}{\sqrt [7]{b}}-\frac {\sin \left (\frac {\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )+b^{2/7} x^2\right )}{2 \sqrt [7]{b}}\right )}{7 a^{6/7}}+\frac {2 \left (\frac {\sin \left (\frac {3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )+b^{2/7} x^2\right )}{2 \sqrt [7]{b}}+\frac {\cos \left (\frac {3 \pi }{14}\right ) \arctan \left (\frac {\sec \left (\frac {3 \pi }{14}\right ) \left (2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right )+2 b^{2/7} x\right )}{2 \sqrt [7]{a} \sqrt [7]{b}}\right )}{\sqrt [7]{b}}\right )}{7 a^{6/7}}+\frac {2 \left (\frac {\sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {\csc \left (\frac {\pi }{7}\right ) \left (2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right )\right )}{2 \sqrt [7]{a} \sqrt [7]{b}}\right )}{\sqrt [7]{b}}-\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )+b^{2/7} x^2\right )}{2 \sqrt [7]{b}}\right )}{7 a^{6/7}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}\) |
Log[a^(1/7) + b^(1/7)*x]/(7*a^(6/7)*b^(1/7)) + (2*((ArcTan[(Sec[Pi/14]*(2* b^(2/7)*x - 2*a^(1/7)*b^(1/7)*Sin[Pi/14]))/(2*a^(1/7)*b^(1/7))]*Cos[Pi/14] )/b^(1/7) - (Log[a^(2/7) + b^(2/7)*x^2 - 2*a^(1/7)*b^(1/7)*x*Sin[Pi/14]]*S in[Pi/14])/(2*b^(1/7))))/(7*a^(6/7)) + (2*(-1/2*(Cos[Pi/7]*Log[a^(2/7) + b ^(2/7)*x^2 - 2*a^(1/7)*b^(1/7)*x*Cos[Pi/7]])/b^(1/7) + (ArcTan[((2*b^(2/7) *x - 2*a^(1/7)*b^(1/7)*Cos[Pi/7])*Csc[Pi/7])/(2*a^(1/7)*b^(1/7))]*Sin[Pi/7 ])/b^(1/7)))/(7*a^(6/7)) + (2*((ArcTan[(Sec[(3*Pi)/14]*(2*b^(2/7)*x + 2*a^ (1/7)*b^(1/7)*Sin[(3*Pi)/14]))/(2*a^(1/7)*b^(1/7))]*Cos[(3*Pi)/14])/b^(1/7 ) + (Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Sin[(3*Pi)/14]]*Sin[( 3*Pi)/14])/(2*b^(1/7))))/(7*a^(6/7))
3.15.43.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; r/(a*n) Int[1/(r + s*x), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 1)/2}], x]] /; Fre eQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.89 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.08
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{7}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{6}}}{7 b}\) | \(27\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{7}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{6}}}{7 b}\) | \(27\) |
Exception generated. \[ \int \frac {1}{a+b x^7} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.06 \[ \int \frac {1}{a+b x^7} \, dx=\operatorname {RootSum} {\left (823543 t^{7} a^{6} b - 1, \left ( t \mapsto t \log {\left (7 t a + x \right )} \right )\right )} \]
\[ \int \frac {1}{a+b x^7} \, dx=\int { \frac {1}{b x^{7} + a} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+b x^7} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) \log \left (2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) \log \left (-2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) \log \left (2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) - x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right )}{7 \, a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{7}} \right |}\right )}{7 \, a} \]
1/7*(-a/b)^(1/7)*cos(3/7*pi)*log(2*x*(-a/b)^(1/7)*cos(3/7*pi) + x^2 + (-a/ b)^(2/7))/a - 1/7*(-a/b)^(1/7)*cos(2/7*pi)*log(-2*x*(-a/b)^(1/7)*cos(2/7*p i) + x^2 + (-a/b)^(2/7))/a + 1/7*(-a/b)^(1/7)*cos(1/7*pi)*log(2*x*(-a/b)^( 1/7)*cos(1/7*pi) + x^2 + (-a/b)^(2/7))/a + 2/7*(-a/b)^(1/7)*arctan(((-a/b) ^(1/7)*cos(3/7*pi) + x)/((-a/b)^(1/7)*sin(3/7*pi)))*sin(3/7*pi)/a + 2/7*(- a/b)^(1/7)*arctan(-((-a/b)^(1/7)*cos(2/7*pi) - x)/((-a/b)^(1/7)*sin(2/7*pi )))*sin(2/7*pi)/a + 2/7*(-a/b)^(1/7)*arctan(((-a/b)^(1/7)*cos(1/7*pi) + x) /((-a/b)^(1/7)*sin(1/7*pi)))*sin(1/7*pi)/a - 1/7*(-a/b)^(1/7)*log(abs(x - (-a/b)^(1/7)))/a
Time = 6.70 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.59 \[ \int \frac {1}{a+b x^7} \, dx=\frac {\ln \left (b^{1/7}\,x+a^{1/7}\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}} \]
log(b^(1/7)*x + a^(1/7))/(7*a^(6/7)*b^(1/7)) - (exp((pi*1i)/7)*log(b^(1/7) *x - a^(1/7)*exp((pi*1i)/7)))/(7*a^(6/7)*b^(1/7)) + (exp((pi*2i)/7)*log(a^ (1/7)*exp((pi*2i)/7) + b^(1/7)*x))/(7*a^(6/7)*b^(1/7)) - (exp((pi*3i)/7)*l og(b^(1/7)*x - a^(1/7)*exp((pi*3i)/7)))/(7*a^(6/7)*b^(1/7)) + (exp((pi*4i) /7)*log(a^(1/7)*exp((pi*4i)/7) + b^(1/7)*x))/(7*a^(6/7)*b^(1/7)) - (exp((p i*5i)/7)*log(b^(1/7)*x - a^(1/7)*exp((pi*5i)/7)))/(7*a^(6/7)*b^(1/7)) + (e xp((pi*6i)/7)*log(a^(1/7)*exp((pi*6i)/7) + b^(1/7)*x))/(7*a^(6/7)*b^(1/7))