Integrand size = 27, antiderivative size = 122 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {x}}+\frac {2 \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt [4]{e} \sqrt {d+e x^2}} \] Output:
-2*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(1/2)+2*(-e)^(1/2)*(d^(1/2)+e^(1 /2)*x)*((e*x^2+d)/(d^(1/2)+e^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(e^ (1/4)*x^(1/2)/d^(1/4)),1/2*2^(1/2))/d^(1/4)/e^(1/4)/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.94 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {x}}+\frac {4 i \sqrt {-e} \sqrt {1+\frac {d}{e x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} \sqrt {d+e x^2}} \] Input:
Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(3/2),x]
Output:
(-2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/Sqrt[x] + ((4*I)*Sqrt[-e]*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1 ])/(Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[d + e*x^2])
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5674, 266, 761}
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.
\(\displaystyle \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 5674 |
\(\displaystyle 2 \sqrt {-e} \int \frac {1}{\sqrt {x} \sqrt {e x^2+d}}dx-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {x}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle 4 \sqrt {-e} \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {x}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt [4]{e} \sqrt {d+e x^2}}-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {x}}\) |
Input:
Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(3/2),x]
Output:
(-2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/Sqrt[x] + (2*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[( e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(d^(1/4)*e^(1/4)*Sqrt[d + e*x^2])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
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]
\[\int \frac {\arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{x^{\frac {3}{2}}}d x\]
Input:
int(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(3/2),x)
Output:
int(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(3/2),x)
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {-e} \sqrt {e} x {\rm weierstrassPInverse}\left (-\frac {4 \, d}{e}, 0, x\right ) - e \sqrt {x} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )\right )}}{e x} \] Input:
integrate(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(3/2),x, algorithm="frica s")
Output:
2*(2*sqrt(-e)*sqrt(e)*x*weierstrassPInverse(-4*d/e, 0, x) - e*sqrt(x)*arct an(sqrt(-e)*x/sqrt(e*x^2 + d)))/(e*x)
Result contains complex when optimal does not.
Time = 8.79 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.58 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=- \frac {2 \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{\sqrt {x}} + \frac {\sqrt {x} \sqrt {- e} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{\sqrt {d} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate(atan((-e)**(1/2)*x/(e*x**2+d)**(1/2))/x**(3/2),x)
Output:
-2*atan(x*sqrt(-e)/sqrt(d + e*x**2))/sqrt(x) + sqrt(x)*sqrt(-e)*gamma(1/4) *hyper((1/4, 1/2), (5/4,), e*x**2*exp_polar(I*pi)/d)/(sqrt(d)*gamma(5/4))
\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(3/2),x, algorithm="maxim a")
Output:
2*(d*sqrt(-e)*sqrt(x)*integrate(-sqrt(e*x^2 + d)*x/((e^2*x^4 + d*e*x^2)*x^ (3/2) - (e*x^2 + d)*e^(log(e*x^2 + d) + 3/2*log(x))), x) - arctan2(sqrt(-e )*x, sqrt(e*x^2 + d)))/sqrt(x)
\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(3/2),x, algorithm="giac" )
Output:
integrate(arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x^(3/2), x)
Timed out. \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=\int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^{3/2}} \,d x \] Input:
int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^(3/2),x)
Output:
int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^(3/2), x)
\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx=\int \frac {\sqrt {x}\, \mathit {atan} \left (\frac {\sqrt {e}\, i x}{\sqrt {e \,x^{2}+d}}\right )}{x^{2}}d x \] Input:
int(atan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(3/2),x)
Output:
int((sqrt(x)*atan((sqrt(e)*i*x)/sqrt(d + e*x**2)))/x**2,x)