\(\int \frac {\arctan (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}})}{x^{11/2}} \, dx\) [23]

Optimal result

Integrand size = 27, antiderivative size = 186 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=-\frac {4 \sqrt {-e} \sqrt {d+e x^2}}{63 d x^{7/2}}-\frac {20 (-e)^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {10 \sqrt {-e} e^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{189 d^{9/4} \sqrt {d+e x^2}} \] Output:

-4/63*(-e)^(1/2)*(e*x^2+d)^(1/2)/d/x^(7/2)-20/189*(-e)^(3/2)*(e*x^2+d)^(1/ 
2)/d^2/x^(3/2)-2/9*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(9/2)+10/189*(-e 
)^(1/2)*e^(7/4)*(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^(9/4)/( 
e*x^2+d)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\frac {4 \sqrt {-e} x \sqrt {d+e x^2} \left (-3 d+5 e x^2\right )-42 d^2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{189 d^2 x^{9/2}}+\frac {20 i (-e)^{5/2} \sqrt {1+\frac {d}{e x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right ),-1\right )}{189 d^2 \sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} \sqrt {d+e x^2}} \] Input:

Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(11/2),x]
 

Output:

(4*Sqrt[-e]*x*Sqrt[d + e*x^2]*(-3*d + 5*e*x^2) - 42*d^2*ArcTan[(Sqrt[-e]*x 
)/Sqrt[d + e*x^2]])/(189*d^2*x^(9/2)) + (((20*I)/189)*(-e)^(5/2)*Sqrt[1 + 
d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/ 
(d^2*Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5674, 264, 264, 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.

\(\displaystyle \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx\)

\(\Big \downarrow \) 5674

\(\displaystyle \frac {2}{9} \sqrt {-e} \int \frac {1}{x^{9/2} \sqrt {e x^2+d}}dx-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {2}{9} \sqrt {-e} \left (-\frac {5 e \int \frac {1}{x^{5/2} \sqrt {e x^2+d}}dx}{7 d}-\frac {2 \sqrt {d+e x^2}}{7 d x^{7/2}}\right )-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {2}{9} \sqrt {-e} \left (-\frac {5 e \left (-\frac {e \int \frac {1}{\sqrt {x} \sqrt {e x^2+d}}dx}{3 d}-\frac {2 \sqrt {d+e x^2}}{3 d x^{3/2}}\right )}{7 d}-\frac {2 \sqrt {d+e x^2}}{7 d x^{7/2}}\right )-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2}{9} \sqrt {-e} \left (-\frac {5 e \left (-\frac {2 e \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}}{3 d}-\frac {2 \sqrt {d+e x^2}}{3 d x^{3/2}}\right )}{7 d}-\frac {2 \sqrt {d+e x^2}}{7 d x^{7/2}}\right )-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {2}{9} \sqrt {-e} \left (-\frac {5 e \left (-\frac {e^{3/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{3 d^{5/4} \sqrt {d+e x^2}}-\frac {2 \sqrt {d+e x^2}}{3 d x^{3/2}}\right )}{7 d}-\frac {2 \sqrt {d+e x^2}}{7 d x^{7/2}}\right )-\frac {2 \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}\)

Input:

Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(11/2),x]
 

Output:

(-2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(9*x^(9/2)) + (2*Sqrt[-e]*((-2*S 
qrt[d + e*x^2])/(7*d*x^(7/2)) - (5*e*((-2*Sqrt[d + e*x^2])/(3*d*x^(3/2)) - 
 (e^(3/4)*(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])/(3*d^(5/4)*Sqrt[d + e 
*x^2])))/(7*d)))/9
 

Defintions of rubi rules used

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 266

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]
 

rule 761

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]
 

rule 5674

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]
 

Maple [F]

\[\int \frac {\arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{x^{\frac {11}{2}}}d x\]

Input:

int(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(11/2),x)
 

Output:

int(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(11/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.46 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\frac {2 \, {\left (10 \, \sqrt {-e} e^{\frac {3}{2}} x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, d}{e}, 0, x\right ) - 21 \, d^{2} \sqrt {x} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + 2 \, {\left (5 \, e x^{3} - 3 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {-e} \sqrt {x}\right )}}{189 \, d^{2} x^{5}} \] Input:

integrate(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(11/2),x, algorithm="fric 
as")
 

Output:

2/189*(10*sqrt(-e)*e^(3/2)*x^5*weierstrassPInverse(-4*d/e, 0, x) - 21*d^2* 
sqrt(x)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + 2*(5*e*x^3 - 3*d*x)*sqrt(e*x^ 
2 + d)*sqrt(-e)*sqrt(x))/(d^2*x^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\text {Timed out} \] Input:

integrate(atan((-e)**(1/2)*x/(e*x**2+d)**(1/2))/x**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {11}{2}}} \,d x } \] Input:

integrate(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(11/2),x, algorithm="maxi 
ma")
 

Output:

2/9*(9*d*sqrt(-e)*x^(9/2)*integrate(-1/9*sqrt(e*x^2 + d)*x/((e^2*x^4 + d*e 
*x^2)*x^(11/2) - (e*x^2 + d)*e^(log(e*x^2 + d) + 11/2*log(x))), x) - arcta 
n2(sqrt(-e)*x, sqrt(e*x^2 + d)))/x^(9/2)
 

Giac [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {11}{2}}} \,d x } \] Input:

integrate(arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(11/2),x, algorithm="giac 
")
 

Output:

integrate(arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x^(11/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^{11/2}} \,d x \] Input:

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^(11/2),x)
 

Output:

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^(11/2), x)
 

Reduce [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx=\int \frac {\sqrt {x}\, \mathit {atan} \left (\frac {\sqrt {e}\, i x}{\sqrt {e \,x^{2}+d}}\right )}{x^{6}}d x \] Input:

int(atan((-e)^(1/2)*x/(e*x^2+d)^(1/2))/x^(11/2),x)
 

Output:

int((sqrt(x)*atan((sqrt(e)*i*x)/sqrt(d + e*x**2)))/x**6,x)