Integrand size = 25, antiderivative size = 116 \[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {3 d x \sqrt {d+e x^2}}{32 (-e)^{3/2}}+\frac {x^3 \sqrt {d+e x^2}}{16 \sqrt {-e}}+\frac {1}{4} x^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {3 d^2 \sqrt {-e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 e^{5/2}} \]
1/4*x^4*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))-3/32*d^2*arctanh(x*e^(1/2)/(e *x^2+d)^(1/2))*(-e)^(1/2)/e^(5/2)+3/32*d*x*(e*x^2+d)^(1/2)/(-e)^(3/2)+1/16 *x^3*(e*x^2+d)^(1/2)/(-e)^(1/2)
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {\sqrt {-e} x \left (3 d-2 e x^2\right ) \sqrt {d+e x^2}+\left (-3 d^2+8 e^2 x^4\right ) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{32 e^2} \]
(Sqrt[-e]*x*(3*d - 2*e*x^2)*Sqrt[d + e*x^2] + (-3*d^2 + 8*e^2*x^4)*ArcTan[ (Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(32*e^2)
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5674, 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.
\(\displaystyle \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle \frac {1}{4} x^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{4} \sqrt {-e} \int \frac {x^4}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{4} x^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{4} \sqrt {-e} \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \int \frac {x^2}{\sqrt {e x^2+d}}dx}{4 e}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{4} x^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{4} \sqrt {-e} \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{\sqrt {e x^2+d}}dx}{2 e}\right )}{4 e}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{4} x^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{4} \sqrt {-e} \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 e}\right )}{4 e}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} x^4 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{4} \sqrt {-e} \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}\right )}{4 e}\right )\) |
(x^4*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/4 - (Sqrt[-e]*((x^3*Sqrt[d + e* x^2])/(4*e) - (3*d*((x*Sqrt[d + e*x^2])/(2*e) - (d*ArcTanh[(Sqrt[e]*x)/Sqr t[d + e*x^2]])/(2*e^(3/2))))/(4*e)))/4
3.1.4.3.1 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. \(211\) vs. \(2(88)=176\).
Time = 0.02 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.83
method | result | size |
default | \(\frac {x^{4} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{4}+\frac {\sqrt {-e}\, e \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{4 d}-\frac {\sqrt {-e}\, \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{4 d}\) | \(212\) |
parts | \(\frac {x^{4} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{4}+\frac {\sqrt {-e}\, e \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{4 d}-\frac {\sqrt {-e}\, \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{4 d}\) | \(212\) |
1/4*x^4*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+1/4*(-e)^(1/2)*e/d*(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)))))-1/4*(-e)^(1/2)/d *(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 = 65, normalized size of antiderivative = 0.56 \[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=-\frac {{\left (2 \, e x^{3} - 3 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {-e} - {\left (8 \, e^{2} x^{4} - 3 \, d^{2}\right )} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{32 \, e^{2}} \]
-1/32*((2*e*x^3 - 3*d*x)*sqrt(e*x^2 + d)*sqrt(-e) - (8*e^2*x^4 - 3*d^2)*ar ctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/e^2
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.92 \[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} - \frac {3 d^{2} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{32 e^{2}} + \frac {3 d x \sqrt {- e} \sqrt {d + e x^{2}}}{32 e^{2}} + \frac {x^{4} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{4} - \frac {x^{3} \sqrt {- e} \sqrt {d + e x^{2}}}{16 e} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-3*d**2*atan(x*sqrt(-e)/sqrt(d + e*x**2))/(32*e**2) + 3*d*x*sqr t(-e)*sqrt(d + e*x**2)/(32*e**2) + x**4*atan(x*sqrt(-e)/sqrt(d + e*x**2))/ 4 - x**3*sqrt(-e)*sqrt(d + e*x**2)/(16*e), Ne(e, 0)), (0, True))
\[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
1/4*x^4*arctan2(sqrt(-e)*x, sqrt(e*x^2 + d)) - d*sqrt(-e)*integrate(-1/4*s qrt(e*x^2 + d)*x^4/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x)
Time = 0.35 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.69 \[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{4} \, x^{4} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {1}{32} \, \sqrt {-e^{2} x^{2} - d e} x {\left (\frac {2 \, x^{2}}{e} - \frac {3 \, d}{e^{2}}\right )} + \frac {3 \, d^{2} \arcsin \left (\frac {e x}{\sqrt {-d e}}\right ) \mathrm {sgn}\left (e\right )}{32 \, e {\left | e \right |}} \]
1/4*x^4*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 1/32*sqrt(-e^2*x^2 - d*e)*x*( 2*x^2/e - 3*d/e^2) + 3/32*d^2*arcsin(e*x/sqrt(-d*e))*sgn(e)/(e*abs(e))
Timed out. \[ \int x^3 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^3\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]