Integrand size = 25, antiderivative size = 124 \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {d^3 \sqrt {d+e x^2}}{7 (-e)^{7/2}}-\frac {d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac {3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}-\frac {\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac {1}{7} x^7 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
-1/7*d^2*(e*x^2+d)^(3/2)/(-e)^(7/2)+3/35*d*(e*x^2+d)^(5/2)/(-e)^(7/2)-1/49 *(e*x^2+d)^(7/2)/(-e)^(7/2)+1/7*x^7*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+1 /7*d^3*(e*x^2+d)^(1/2)/(-e)^(7/2)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67 \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {\sqrt {d+e x^2} \left (16 d^3-8 d^2 e x^2+6 d e^2 x^4-5 e^3 x^6\right )}{245 (-e)^{7/2}}+\frac {1}{7} x^7 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
(Sqrt[d + e*x^2]*(16*d^3 - 8*d^2*e*x^2 + 6*d*e^2*x^4 - 5*e^3*x^6))/(245*(- e)^(7/2)) + (x^7*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7
Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5674, 243, 53, 2009}
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.
\(\displaystyle \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle \frac {1}{7} x^7 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{7} \sqrt {-e} \int \frac {x^7}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{7} x^7 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{14} \sqrt {-e} \int \frac {x^6}{\sqrt {e x^2+d}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{7} x^7 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{14} \sqrt {-e} \int \left (-\frac {d^3}{e^3 \sqrt {e x^2+d}}+\frac {3 \sqrt {e x^2+d} d^2}{e^3}-\frac {3 \left (e x^2+d\right )^{3/2} d}{e^3}+\frac {\left (e x^2+d\right )^{5/2}}{e^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} x^7 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{14} \sqrt {-e} \left (-\frac {2 d^3 \sqrt {d+e x^2}}{e^4}+\frac {2 d^2 \left (d+e x^2\right )^{3/2}}{e^4}+\frac {2 \left (d+e x^2\right )^{7/2}}{7 e^4}-\frac {6 d \left (d+e x^2\right )^{5/2}}{5 e^4}\right )\) |
-1/14*(Sqrt[-e]*((-2*d^3*Sqrt[d + e*x^2])/e^4 + (2*d^2*(d + e*x^2)^(3/2))/ e^4 - (6*d*(d + e*x^2)^(5/2))/(5*e^4) + (2*(d + e*x^2)^(7/2))/(7*e^4))) + (x^7*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7
3.1.11.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(94)=188\).
Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {x^{7} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{7}-\frac {\sqrt {-e}\, \left (\frac {x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{9 e}-\frac {2 d \left (\frac {x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 e}-\frac {4 d \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{7 e}\right )}{3 e}\right )}{7 d}+\frac {\sqrt {-e}\, e \left (\frac {x^{8} \sqrt {e \,x^{2}+d}}{9 e}-\frac {8 d \left (\frac {x^{6} \sqrt {e \,x^{2}+d}}{7 e}-\frac {6 d \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{7 e}\right )}{9 e}\right )}{7 d}\) | \(231\) |
parts | \(\frac {x^{7} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{7}-\frac {\sqrt {-e}\, \left (\frac {x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{9 e}-\frac {2 d \left (\frac {x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 e}-\frac {4 d \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{7 e}\right )}{3 e}\right )}{7 d}+\frac {\sqrt {-e}\, e \left (\frac {x^{8} \sqrt {e \,x^{2}+d}}{9 e}-\frac {8 d \left (\frac {x^{6} \sqrt {e \,x^{2}+d}}{7 e}-\frac {6 d \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{7 e}\right )}{9 e}\right )}{7 d}\) | \(231\) |
1/7*x^7*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))-1/7*(-e)^(1/2)/d*(1/9*x^6*(e* x^2+d)^(3/2)/e-2/3*d/e*(1/7*x^4*(e*x^2+d)^(3/2)/e-4/7*d/e*(1/5*x^2*(e*x^2+ d)^(3/2)/e-2/15*d/e^2*(e*x^2+d)^(3/2))))+1/7*(-e)^(1/2)*e/d*(1/9*x^8/e*(e* x^2+d)^(1/2)-8/9*d/e*(1/7*x^6/e*(e*x^2+d)^(1/2)-6/7*d/e*(1/5*x^4/e*(e*x^2+ d)^(1/2)-4/5*d/e*(1/3*x^2/e*(e*x^2+d)^(1/2)-2/3*d/e^2*(e*x^2+d)^(1/2)))))
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {35 \, e^{4} x^{7} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (5 \, e^{3} x^{6} - 6 \, d e^{2} x^{4} + 8 \, d^{2} e x^{2} - 16 \, d^{3}\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{245 \, e^{4}} \]
1/245*(35*e^4*x^7*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (5*e^3*x^6 - 6*d*e^ 2*x^4 + 8*d^2*e*x^2 - 16*d^3)*sqrt(e*x^2 + d)*sqrt(-e))/e^4
Time = 1.43 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} \frac {16 d^{3} \sqrt {- e} \sqrt {d + e x^{2}}}{245 e^{4}} - \frac {8 d^{2} x^{2} \sqrt {- e} \sqrt {d + e x^{2}}}{245 e^{3}} + \frac {6 d x^{4} \sqrt {- e} \sqrt {d + e x^{2}}}{245 e^{2}} + \frac {x^{7} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{7} - \frac {x^{6} \sqrt {- e} \sqrt {d + e x^{2}}}{49 e} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((16*d**3*sqrt(-e)*sqrt(d + e*x**2)/(245*e**4) - 8*d**2*x**2*sqrt (-e)*sqrt(d + e*x**2)/(245*e**3) + 6*d*x**4*sqrt(-e)*sqrt(d + e*x**2)/(245 *e**2) + x**7*atan(x*sqrt(-e)/sqrt(d + e*x**2))/7 - x**6*sqrt(-e)*sqrt(d + e*x**2)/(49*e), Ne(e, 0)), (0, True))
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.35 \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{7} \, x^{7} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {{\left (35 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 135 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 189 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} - 105 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3}\right )} \sqrt {-e}}{2205 \, d e^{4}} + \frac {{\left (35 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x^{2} + d} d^{4}\right )} \sqrt {-e}}{2205 \, d e^{4}} \]
1/7*x^7*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 1/2205*(35*(e*x^2 + d)^(9/2) - 135*(e*x^2 + d)^(7/2)*d + 189*(e*x^2 + d)^(5/2)*d^2 - 105*(e*x^2 + d)^(3 /2)*d^3)*sqrt(-e)/(d*e^4) + 1/2205*(35*(e*x^2 + d)^(9/2) - 180*(e*x^2 + d) ^(7/2)*d + 378*(e*x^2 + d)^(5/2)*d^2 - 420*(e*x^2 + d)^(3/2)*d^3 + 315*sqr t(e*x^2 + d)*d^4)*sqrt(-e)/(d*e^4)
Time = 0.32 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{7} \, x^{7} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + \frac {\sqrt {-e^{2} x^{2} - d e} d^{3}}{7 \, e^{4}} + \frac {35 \, {\left (-e^{2} x^{2} - d e\right )}^{\frac {3}{2}} d^{2} e^{2} + 21 \, {\left (e^{2} x^{2} + d e\right )}^{2} \sqrt {-e^{2} x^{2} - d e} d e - 5 \, {\left (e^{2} x^{2} + d e\right )}^{3} \sqrt {-e^{2} x^{2} - d e}}{245 \, e^{7}} \]
1/7*x^7*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + 1/7*sqrt(-e^2*x^2 - d*e)*d^3/ e^4 + 1/245*(35*(-e^2*x^2 - d*e)^(3/2)*d^2*e^2 + 21*(e^2*x^2 + d*e)^2*sqrt (-e^2*x^2 - d*e)*d*e - 5*(e^2*x^2 + d*e)^3*sqrt(-e^2*x^2 - d*e))/e^7
Timed out. \[ \int x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^6\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]