Integrand size = 23, antiderivative size = 88 \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {x \sqrt {d+e x^2}}{4 \sqrt {-e}}+\frac {1}{2} x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {d \sqrt {-e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 e^{3/2}} \] Output:
1/4*x*(e*x^2+d)^(1/2)/(-e)^(1/2)+1/2*x^2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/ 2))+1/4*d*(-e)^(1/2)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/e^(3/2)
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.67 \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {-\sqrt {-e} x \sqrt {d+e x^2}+\left (d+2 e x^2\right ) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{4 e} \] Input:
Integrate[x*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]
Output:
(-(Sqrt[-e]*x*Sqrt[d + e*x^2]) + (d + 2*e*x^2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(4*e)
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5674, 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.
\(\displaystyle \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle \frac {1}{2} x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{2} \sqrt {-e} \int \frac {x^2}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{2} \sqrt {-e} \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{\sqrt {e x^2+d}}dx}{2 e}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{2} \sqrt {-e} \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 e}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} x^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{2} \sqrt {-e} \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}\right )\) |
Input:
Int[x*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]
Output:
(x^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/2 - (Sqrt[-e]*((x*Sqrt[d + e*x^ 2])/(2*e) - (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*e^(3/2))))/2
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. \(163\) vs. \(2(66)=132\).
Time = 0.02 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {x^{2} \arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{2}+\frac {\sqrt {-e}\, e \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{2 d}-\frac {\sqrt {-e}\, \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 d}\) | \(164\) |
parts | \(\frac {x^{2} \arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{2}+\frac {\sqrt {-e}\, e \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{2 d}-\frac {\sqrt {-e}\, \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 d}\) | \(164\) |
Input:
int(x*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/2*x^2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))+1/2*(-e)^(1/2)*e/d*(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(e^(1/2) *x+(e*x^2+d)^(1/2))))-1/2*(-e)^(1/2)/d*(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(e^(1/2)*x+(e*x^2+d)^(1/2))))
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=-\frac {\sqrt {e x^{2} + d} \sqrt {-e} x - {\left (2 \, e x^{2} + d\right )} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{4 \, e} \] Input:
integrate(x*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="fricas")
Output:
-1/4*(sqrt(e*x^2 + d)*sqrt(-e)*x - (2*e*x^2 + d)*arctan(sqrt(-e)*x/sqrt(e* x^2 + d)))/e
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83 \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} \frac {d \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{4 e} + \frac {x^{2} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{2} - \frac {x \sqrt {- e} \sqrt {d + e x^{2}}}{4 e} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*atan((-e)**(1/2)*x/(e*x**2+d)**(1/2)),x)
Output:
Piecewise((d*atan(x*sqrt(-e)/sqrt(d + e*x**2))/(4*e) + x**2*atan(x*sqrt(-e )/sqrt(d + e*x**2))/2 - x*sqrt(-e)*sqrt(d + e*x**2)/(4*e), Ne(e, 0)), (0, True))
\[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \] Input:
integrate(x*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="maxima")
Output:
1/2*x^2*arctan2(sqrt(-e)*x, sqrt(e*x^2 + d)) - d*sqrt(-e)*integrate(-1/2*s qrt(e*x^2 + d)*x^2/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x)
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{2} \, x^{2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {d \arcsin \left (\frac {e x}{\sqrt {-d e}}\right ) \mathrm {sgn}\left (e\right )}{4 \, {\left | e \right |}} - \frac {\sqrt {-e^{2} x^{2} - d e} x}{4 \, e} \] Input:
integrate(x*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="giac")
Output:
1/2*x^2*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 1/4*d*arcsin(e*x/sqrt(-d*e))* sgn(e)/abs(e) - 1/4*sqrt(-e^2*x^2 - d*e)*x/e
Timed out. \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \] Input:
int(x*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)),x)
Output:
int(x*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)), x)
Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.57 \[ \int x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {2 \mathit {atan} \left (\frac {\sqrt {e}\, \sqrt {e \,x^{2}+d}\, i x +e i \,x^{2}}{\sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +d +e \,x^{2}}\right ) d +2 \mathit {atan} \left (\frac {\sqrt {e}\, \sqrt {e \,x^{2}+d}\, i x +e i \,x^{2}}{\sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +d +e \,x^{2}}\right ) e \,x^{2}-\sqrt {e}\, \sqrt {e \,x^{2}+d}\, i x -\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d i}{4 e} \] Input:
int(x*atan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x)
Output:
(2*atan((sqrt(e)*sqrt(d + e*x**2)*i*x + e*i*x**2)/(sqrt(e)*sqrt(d + e*x**2 )*x + d + e*x**2))*d + 2*atan((sqrt(e)*sqrt(d + e*x**2)*i*x + e*i*x**2)/(s qrt(e)*sqrt(d + e*x**2)*x + d + e*x**2))*e*x**2 - sqrt(e)*sqrt(d + e*x**2) *i*x - log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d*i)/(4*e)