Integrand size = 25, antiderivative size = 113 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{30 d x^5}-\frac {2 (-e)^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {4 (-e)^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6} \]
-1/6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^6-2/45*(-e)^(3/2)*(e*x^2+d)^(1 /2)/d^2/x^3-4/45*(-e)^(5/2)*(e*x^2+d)^(1/2)/d^3/x-1/30*(-e)^(1/2)*(e*x^2+d )^(1/2)/d/x^5
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.69 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\frac {\sqrt {-e} x \sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )-15 d^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{90 d^3 x^6} \]
(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-3*d^2 + 4*d*e*x^2 - 8*e^2*x^4) - 15*d^3*ArcT an[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(90*d^3*x^6)
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5674, 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.
\(\displaystyle \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle \frac {1}{6} \sqrt {-e} \int \frac {1}{x^6 \sqrt {e x^2+d}}dx-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {1}{6} \sqrt {-e} \left (-\frac {4 e \int \frac {1}{x^4 \sqrt {e x^2+d}}dx}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {1}{6} \sqrt {-e} \left (-\frac {4 e \left (-\frac {2 e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle \frac {1}{6} \sqrt {-e} \left (-\frac {4 e \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )-\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
(Sqrt[-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)))/6 - ArcTan[(Sqrt[-e]*x)/Sqr t[d + e*x^2]]/(6*x^6)
3.1.9.3.1 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 = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}-\frac {\sqrt {-e}\, e \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{6 d}+\frac {\sqrt {-e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{6 d}\) | \(117\) |
parts | \(-\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}-\frac {\sqrt {-e}\, e \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{6 d}+\frac {\sqrt {-e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{6 d}\) | \(117\) |
-1/6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^6-1/6*(-e)^(1/2)*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))+1/6*(-e)^(1/2)/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 = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.60 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=-\frac {15 \, d^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (8 \, e^{2} x^{5} - 4 \, d e x^{3} + 3 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{90 \, d^{3} x^{6}} \]
-1/90*(15*d^3*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (8*e^2*x^5 - 4*d*e*x^3 + 3*d^2*x)*sqrt(e*x^2 + d)*sqrt(-e))/(d^3*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (102) = 204\).
Time = 2.68 (sec) , antiderivative size = 352, normalized size of antiderivative = 3.12 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=- \frac {d^{4} e^{\frac {9}{2}} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{30 d^{5} e^{4} x^{4} + 60 d^{4} e^{5} x^{6} + 30 d^{3} e^{6} x^{8}} - \frac {d^{3} e^{\frac {11}{2}} x^{2} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{45 d^{5} e^{4} x^{4} + 90 d^{4} e^{5} x^{6} + 45 d^{3} e^{6} x^{8}} - \frac {d^{2} e^{\frac {13}{2}} x^{4} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{30 d^{5} e^{4} x^{4} + 60 d^{4} e^{5} x^{6} + 30 d^{3} e^{6} x^{8}} - \frac {2 d e^{\frac {15}{2}} x^{6} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}} - \frac {4 e^{\frac {17}{2}} x^{8} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{45 d^{5} e^{4} x^{4} + 90 d^{4} e^{5} x^{6} + 45 d^{3} e^{6} x^{8}} - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{6 x^{6}} \]
-d**4*e**(9/2)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(30*d**5*e**4*x**4 + 60*d**4* e**5*x**6 + 30*d**3*e**6*x**8) - d**3*e**(11/2)*x**2*sqrt(-e)*sqrt(d/(e*x* *2) + 1)/(45*d**5*e**4*x**4 + 90*d**4*e**5*x**6 + 45*d**3*e**6*x**8) - d** 2*e**(13/2)*x**4*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(30*d**5*e**4*x**4 + 60*d** 4*e**5*x**6 + 30*d**3*e**6*x**8) - 2*d*e**(15/2)*x**6*sqrt(-e)*sqrt(d/(e*x **2) + 1)/(15*d**5*e**4*x**4 + 30*d**4*e**5*x**6 + 15*d**3*e**6*x**8) - 4* e**(17/2)*x**8*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(45*d**5*e**4*x**4 + 90*d**4* e**5*x**6 + 45*d**3*e**6*x**8) - atan(x*sqrt(-e)/sqrt(d + e*x**2))/(6*x**6 )
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=-\frac {{\left (2 \, e^{2} x^{4} + d e x^{2} - d^{2}\right )} \sqrt {-e} e}{18 \, \sqrt {e x^{2} + d} d^{3} x^{3}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{6 \, x^{6}} + \frac {{\left (2 \, e^{2} x^{4} - d e x^{2} - 3 \, d^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{90 \, d^{3} x^{5}} \]
-1/18*(2*e^2*x^4 + d*e*x^2 - d^2)*sqrt(-e)*e/(sqrt(e*x^2 + d)*d^3*x^3) - 1 /6*arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x^6 + 1/90*(2*e^2*x^4 - d*e*x^2 - 3* d^2)*sqrt(e*x^2 + d)*sqrt(-e)/(d^3*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (89) = 178\).
Time = 0.35 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.43 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\frac {{\left (3 \, e^{4} + \frac {25 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{2}}{x^{2}} + \frac {150 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{4}}{e^{4} x^{4}}\right )} e^{10} x^{5}}{2880 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{5} d^{3} {\left | e \right |}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{6 \, x^{6}} - \frac {\frac {150 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )} d^{12} e^{16}}{x} + \frac {25 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{3} d^{12} e^{12}}{x^{3}} + \frac {3 \, {\left (\sqrt {-d e} e + \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )}^{5} d^{12} e^{8}}{x^{5}}}{2880 \, d^{15} e^{14} {\left | e \right |}} \]
1/2880*(3*e^4 + 25*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^2/x^2 + 15 0*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^4/(e^4*x^4))*e^10*x^5/((sqr t(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^5*d^3*abs(e)) - 1/6*arctan(sqrt(- e)*x/sqrt(e*x^2 + d))/x^6 - 1/2880*(150*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d* e)*abs(e))*d^12*e^16/x + 25*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^3 *d^12*e^12/x^3 + 3*(sqrt(-d*e)*e + sqrt(-e^2*x^2 - d*e)*abs(e))^5*d^12*e^8 /x^5)/(d^15*e^14*abs(e))
Timed out. \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^7} \,d x \]