Integrand size = 25, antiderivative size = 144 \[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {5 d^2 x \sqrt {d+e x^2}}{96 (-e)^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}}+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {5 d^3 \sqrt {-e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^{7/2}} \]
1/6*x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+5/96*d^3*arctanh(x*e^(1/2)/(e *x^2+d)^(1/2))*(-e)^(1/2)/e^(7/2)+5/96*d^2*x*(e*x^2+d)^(1/2)/(-e)^(5/2)+5/ 144*d*x^3*(e*x^2+d)^(1/2)/(-e)^(3/2)+1/36*x^5*(e*x^2+d)^(1/2)/(-e)^(1/2)
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.60 \[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {\sqrt {-e} x \sqrt {d+e x^2} \left (-15 d^2+10 d e x^2-8 e^2 x^4\right )+3 \left (5 d^3+16 e^3 x^6\right ) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{288 e^3} \]
(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-15*d^2 + 10*d*e*x^2 - 8*e^2*x^4) + 3*(5*d^3 + 16*e^3*x^6)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(288*e^3)
Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {5674, 262, 262, 262, 224, 219}
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.
\(\displaystyle \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle \frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \int \frac {x^6}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \int \frac {x^4}{\sqrt {e x^2+d}}dx}{6 e}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \int \frac {x^2}{\sqrt {e x^2+d}}dx}{4 e}\right )}{6 e}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{\sqrt {e x^2+d}}dx}{2 e}\right )}{4 e}\right )}{6 e}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 e}\right )}{4 e}\right )}{6 e}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{6} x^6 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}\right )}{4 e}\right )}{6 e}\right )\) |
(x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/6 - (Sqrt[-e]*((x^5*Sqrt[d + e* x^2])/(6*e) - (5*d*((x^3*Sqrt[d + e*x^2])/(4*e) - (3*d*((x*Sqrt[d + e*x^2] )/(2*e) - (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*e^(3/2))))/(4*e)))/( 6*e)))/6
3.1.3.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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. \(259\) vs. \(2(110)=220\).
Time = 0.03 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {x^{6} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{6}+\frac {\sqrt {-e}\, e \left (\frac {x^{7} \sqrt {e \,x^{2}+d}}{8 e}-\frac {7 d \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{8 e}\right )}{6 d}-\frac {\sqrt {-e}\, \left (\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 e}-\frac {5 d \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{8 e}\right )}{6 d}\) | \(260\) |
parts | \(\frac {x^{6} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{6}+\frac {\sqrt {-e}\, e \left (\frac {x^{7} \sqrt {e \,x^{2}+d}}{8 e}-\frac {7 d \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{8 e}\right )}{6 d}-\frac {\sqrt {-e}\, \left (\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 e}-\frac {5 d \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{8 e}\right )}{6 d}\) | \(260\) |
1/6*x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+1/6*(-e)^(1/2)*e/d*(1/8*x^7/e *(e*x^2+d)^(1/2)-7/8*d/e*(1/6*x^5/e*(e*x^2+d)^(1/2)-5/6*d/e*(1/4*x^3/e*(e* x^2+d)^(1/2)-3/4*d/e*(1/2*x/e*(e*x^2+d)^(1/2)-1/2*d/e^(3/2)*ln(x*e^(1/2)+( e*x^2+d)^(1/2))))))-1/6*(-e)^(1/2)/d*(1/8*x^5*(e*x^2+d)^(3/2)/e-5/8*d/e*(1 /6*x^3*(e*x^2+d)^(3/2)/e-1/2*d/e*(1/4*x*(e*x^2+d)^(3/2)/e-1/4*d/e*(1/2*x*( e*x^2+d)^(1/2)+1/2*d/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))))))
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.53 \[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=-\frac {{\left (8 \, e^{2} x^{5} - 10 \, d e x^{3} + 15 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {-e} - 3 \, {\left (16 \, e^{3} x^{6} + 5 \, d^{3}\right )} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{288 \, e^{3}} \]
-1/288*((8*e^2*x^5 - 10*d*e*x^3 + 15*d^2*x)*sqrt(e*x^2 + d)*sqrt(-e) - 3*( 16*e^3*x^6 + 5*d^3)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/e^3
Time = 0.89 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.96 \[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} \frac {5 d^{3} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{96 e^{3}} - \frac {5 d^{2} x \sqrt {- e} \sqrt {d + e x^{2}}}{96 e^{3}} + \frac {5 d x^{3} \sqrt {- e} \sqrt {d + e x^{2}}}{144 e^{2}} + \frac {x^{6} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{6} - \frac {x^{5} \sqrt {- e} \sqrt {d + e x^{2}}}{36 e} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((5*d**3*atan(x*sqrt(-e)/sqrt(d + e*x**2))/(96*e**3) - 5*d**2*x*s qrt(-e)*sqrt(d + e*x**2)/(96*e**3) + 5*d*x**3*sqrt(-e)*sqrt(d + e*x**2)/(1 44*e**2) + x**6*atan(x*sqrt(-e)/sqrt(d + e*x**2))/6 - x**5*sqrt(-e)*sqrt(d + e*x**2)/(36*e), Ne(e, 0)), (0, True))
\[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{5} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
1/6*x^6*arctan2(sqrt(-e)*x, sqrt(e*x^2 + d)) - d*sqrt(-e)*integrate(-1/6*s qrt(e*x^2 + d)*x^6/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x)
Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{6} \, x^{6} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {1}{288} \, \sqrt {-e^{2} x^{2} - d e} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{e} - \frac {5 \, d}{e^{2}}\right )} + \frac {15 \, d^{2}}{e^{3}}\right )} x - \frac {5 \, d^{3} \arcsin \left (\frac {e x}{\sqrt {-d e}}\right ) \mathrm {sgn}\left (e\right )}{96 \, e^{2} {\left | e \right |}} \]
1/6*x^6*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 1/288*sqrt(-e^2*x^2 - d*e)*(2 *x^2*(4*x^2/e - 5*d/e^2) + 15*d^2/e^3)*x - 5/96*d^3*arcsin(e*x/sqrt(-d*e)) *sgn(e)/(e^2*abs(e))
Timed out. \[ \int x^5 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^5\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]