Integrand size = 25, antiderivative size = 141 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{56 d x^7}-\frac {3 (-e)^{3/2} \sqrt {d+e x^2}}{140 d^2 x^5}-\frac {(-e)^{5/2} \sqrt {d+e x^2}}{35 d^3 x^3}-\frac {2 (-e)^{7/2} \sqrt {d+e x^2}}{35 d^4 x}-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{8 x^8} \]
-1/8*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^8-3/140*(-e)^(3/2)*(e*x^2+d)^( 1/2)/d^2/x^5-1/35*(-e)^(5/2)*(e*x^2+d)^(1/2)/d^3/x^3-2/35*(-e)^(7/2)*(e*x^ 2+d)^(1/2)/d^4/x-1/56*(-e)^(1/2)*(e*x^2+d)^(1/2)/d/x^7
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=\frac {\sqrt {-e} x \sqrt {d+e x^2} \left (-5 d^3+6 d^2 e x^2-8 d e^2 x^4+16 e^3 x^6\right )-35 d^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{280 d^4 x^8} \]
(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-5*d^3 + 6*d^2*e*x^2 - 8*d*e^2*x^4 + 16*e^3*x ^6) - 35*d^4*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(280*d^4*x^8)
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5674, 245, 245, 245, 242}
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.
| \(\displaystyle \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 5674 |
| \(\displaystyle \frac {1}{8} \sqrt {-e} \int \frac {1}{x^8 \sqrt {e x^2+d}}dx-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{8 x^8}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {1}{8} \sqrt {-e} \left (-\frac {6 e \int \frac {1}{x^6 \sqrt {e x^2+d}}dx}{7 d}-\frac {\sqrt {d+e x^2}}{7 d x^7}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{8 x^8}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {1}{8} \sqrt {-e} \left (-\frac {6 e \left (-\frac {4 e \int \frac {1}{x^4 \sqrt {e x^2+d}}dx}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{7 d}-\frac {\sqrt {d+e x^2}}{7 d x^7}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{8 x^8}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {1}{8} \sqrt {-e} \left (-\frac {6 e \left (-\frac {4 e \left (-\frac {2 e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{7 d}-\frac {\sqrt {d+e x^2}}{7 d x^7}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{8 x^8}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {1}{8} \sqrt {-e} \left (-\frac {6 e \left (-\frac {4 e \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{7 d}-\frac {\sqrt {d+e x^2}}{7 d x^7}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{8 x^8}\) |
(Sqrt[-e]*(-1/7*Sqrt[d + e*x^2]/(d*x^7) - (6*e*(-1/5*Sqrt[d + e*x^2]/(d*x^ 5) - (4*e*(-1/3*Sqrt[d + e*x^2]/(d*x^3) + (2*e*Sqrt[d + e*x^2])/(3*d^2*x)) )/(5*d)))/(7*d)))/8 - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(8*x^8)
3.1.10.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d*x)^(m + 1)*(ArcTan[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1))), x ] - Simp[c/(d*(m + 1)) Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; FreeQ [{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(-\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{8 x^{8}}-\frac {\sqrt {-e}\, e \left (-\frac {\sqrt {e \,x^{2}+d}}{5 d \,x^{5}}-\frac {4 e \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{5 d}\right )}{8 d}+\frac {\sqrt {-e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 d \,x^{7}}-\frac {4 e \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{7 d}\right )}{8 d}\) | \(165\) |
| parts | \(-\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{8 x^{8}}-\frac {\sqrt {-e}\, e \left (-\frac {\sqrt {e \,x^{2}+d}}{5 d \,x^{5}}-\frac {4 e \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{5 d}\right )}{8 d}+\frac {\sqrt {-e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 d \,x^{7}}-\frac {4 e \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{7 d}\right )}{8 d}\) | \(165\) |
-1/8*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^8-1/8*(-e)^(1/2)*e/d*(-1/5/d/x ^5*(e*x^2+d)^(1/2)-4/5*e/d*(-1/3/d/x^3*(e*x^2+d)^(1/2)+2/3*e/d^2/x*(e*x^2+ d)^(1/2)))+1/8*(-e)^(1/2)/d*(-1/7/d/x^7*(e*x^2+d)^(3/2)-4/7*e/d*(-1/5/d/x^ 5*(e*x^2+d)^(3/2)+2/15*e/d^2/x^3*(e*x^2+d)^(3/2)))
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.57 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=-\frac {35 \, d^{4} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (16 \, e^{3} x^{7} - 8 \, d e^{2} x^{5} + 6 \, d^{2} e x^{3} - 5 \, d^{3} x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{280 \, d^{4} x^{8}} \]
-1/280*(35*d^4*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (16*e^3*x^7 - 8*d*e^2* x^5 + 6*d^2*e*x^3 - 5*d^3*x)*sqrt(e*x^2 + d)*sqrt(-e))/(d^4*x^8)
Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (128) = 256\).
Time = 3.47 (sec) , antiderivative size = 575, normalized size of antiderivative = 4.08 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=- \frac {5 d^{6} e^{\frac {19}{2}} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} - \frac {9 d^{5} e^{\frac {21}{2}} x^{2} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} - \frac {5 d^{4} e^{\frac {23}{2}} x^{4} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} + \frac {5 d^{3} e^{\frac {25}{2}} x^{6} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} + \frac {15 d^{2} e^{\frac {27}{2}} x^{8} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{140 d^{7} e^{9} x^{6} + 420 d^{6} e^{10} x^{8} + 420 d^{5} e^{11} x^{10} + 140 d^{4} e^{12} x^{12}} + \frac {5 d e^{\frac {29}{2}} x^{10} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}} + \frac {2 e^{\frac {31}{2}} x^{12} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}} - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{8 x^{8}} \]
-5*d**6*e**(19/2)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(280*d**7*e**9*x**6 + 840* d**6*e**10*x**8 + 840*d**5*e**11*x**10 + 280*d**4*e**12*x**12) - 9*d**5*e* *(21/2)*x**2*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(280*d**7*e**9*x**6 + 840*d**6* e**10*x**8 + 840*d**5*e**11*x**10 + 280*d**4*e**12*x**12) - 5*d**4*e**(23/ 2)*x**4*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(280*d**7*e**9*x**6 + 840*d**6*e**10 *x**8 + 840*d**5*e**11*x**10 + 280*d**4*e**12*x**12) + 5*d**3*e**(25/2)*x* *6*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(280*d**7*e**9*x**6 + 840*d**6*e**10*x**8 + 840*d**5*e**11*x**10 + 280*d**4*e**12*x**12) + 15*d**2*e**(27/2)*x**8*s qrt(-e)*sqrt(d/(e*x**2) + 1)/(140*d**7*e**9*x**6 + 420*d**6*e**10*x**8 + 4 20*d**5*e**11*x**10 + 140*d**4*e**12*x**12) + 5*d*e**(29/2)*x**10*sqrt(-e) *sqrt(d/(e*x**2) + 1)/(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5* e**11*x**10 + 35*d**4*e**12*x**12) + 2*e**(31/2)*x**12*sqrt(-e)*sqrt(d/(e* x**2) + 1)/(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5*e**11*x**10 + 35*d**4*e**12*x**12) - atan(x*sqrt(-e)/sqrt(d + e*x**2))/(8*x**8)
Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=\frac {{\left (8 \, e^{3} x^{6} + 4 \, d e^{2} x^{4} - d^{2} e x^{2} + 3 \, d^{3}\right )} \sqrt {-e} e}{120 \, \sqrt {e x^{2} + d} d^{4} x^{5}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{8 \, x^{8}} - \frac {{\left (8 \, e^{3} x^{6} - 4 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + 15 \, d^{3}\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{840 \, d^{4} x^{7}} \]
1/120*(8*e^3*x^6 + 4*d*e^2*x^4 - d^2*e*x^2 + 3*d^3)*sqrt(-e)*e/(sqrt(e*x^2 + d)*d^4*x^5) - 1/8*arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x^8 - 1/840*(8*e^3 *x^6 - 4*d*e^2*x^4 + 3*d^2*e*x^2 + 15*d^3)*sqrt(e*x^2 + d)*sqrt(-e)/(d^4*x ^7)
Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (111) = 222\).
Time = 0.35 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.50 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=-\frac {{\left (5 \, e^{5} + \frac {49 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{2} e}{x^{2}} + \frac {245 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{4}}{e^{3} x^{4}} + \frac {1225 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{6}}{e^{7} x^{6}}\right )} e^{14} x^{7}}{35840 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{7} d^{4} {\left | e \right |}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{8 \, x^{8}} + \frac {\frac {1225 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )} d^{24} e^{30}}{x} + \frac {245 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{3} d^{24} e^{26}}{x^{3}} + \frac {49 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{5} d^{24} e^{22}}{x^{5}} + \frac {5 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{7} d^{24} e^{18}}{x^{7}}}{35840 \, d^{28} e^{27} {\left | e \right |}} \]
-1/35840*(5*e^5 + 49*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^2*e/x^2 + 245*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^4/(e^3*x^4) + 1225*(sqr t(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^6/(e^7*x^6))*e^14*x^7/((sqrt(-d*e )*e + sqrt(-e^2*x^2 - d*e)*abs(e))^7*d^4*abs(e)) - 1/8*arctan(sqrt(-e)*x/s qrt(e*x^2 + d))/x^8 + 1/35840*(1225*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*a bs(e))*d^24*e^30/x + 245*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^3*d^ 24*e^26/x^3 + 49*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^5*d^24*e^22/ x^5 + 5*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^7*d^24*e^18/x^7)/(d^2 8*e^27*abs(e))
Timed out. \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx=\int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^9} \,d x \]