\(\int \frac {x \arctan (a x)^{3/2}}{(c+a^2 c x^2)^{3/2}} \, dx\) [823]

Optimal result

Integrand size = 24, antiderivative size = 129 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {3 x \sqrt {\arctan (a x)}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {c+a^2 c x^2}} \] Output:

3/2*x*arctan(a*x)^(1/2)/a/c/(a^2*c*x^2+c)^(1/2)-arctan(a*x)^(3/2)/a^2/c/(a 
^2*c*x^2+c)^(1/2)-3/4*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*FresnelS(2^(1/2)/ 
Pi^(1/2)*arctan(a*x)^(1/2))/a^2/c/(a^2*c*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.99 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {4 (3 a x-2 \arctan (a x)) \arctan (a x)+3 \sqrt {1+a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+3 \sqrt {1+a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )}{8 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \] Input:

Integrate[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]
 

Output:

(4*(3*a*x - 2*ArcTan[a*x])*ArcTan[a*x] + 3*Sqrt[1 + a^2*x^2]*Sqrt[(-I)*Arc 
Tan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan 
[a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(8*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan 
[a*x]])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5465, 5440, 5439, 3042, 3777, 25, 3042, 3786, 3832}

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.

\(\displaystyle \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 c x^2+c\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {3 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{2 a}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5440

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^{3/2}}dx}{2 a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5439

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \int \frac {\sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}d\arctan (a x)}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \int \sqrt {\arctan (a x)} \sin \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{2} \int -\frac {a x}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)+\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\frac {1}{2} \int \frac {a x}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\frac {1}{2} \int \frac {\sin (\arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\int \frac {a x}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\)

Input:

Int[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]
 

Output:

-(ArcTan[a*x]^(3/2)/(a^2*c*Sqrt[c + a^2*c*x^2])) + (3*Sqrt[1 + a^2*x^2]*(( 
a*x*Sqrt[ArcTan[a*x]])/Sqrt[1 + a^2*x^2] - Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]* 
Sqrt[ArcTan[a*x]]]))/(2*a^2*c*Sqrt[c + a^2*c*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 5439

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^q/c   Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar 
cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( 
q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
 

rule 5440

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2])   Int[(1 + 
c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] &&  !(IntegerQ[q] || GtQ[d, 0])
 

rule 5465

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 
1))), x] - Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 
 0] && NeQ[q, -1]
 

Maple [F]

\[\int \frac {x \arctan \left (a x \right )^{\frac {3}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

Input:

int(x*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)
 

Output:

int(x*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x*atan(a*x)**(3/2)/(a**2*c*x**2+c)**(3/2),x)
 

Output:

Integral(x*atan(a*x)**(3/2)/(c*(a**2*x**2 + 1))**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate(x*arctan(a*x)^(3/2)/(a^2*c*x^2 + c)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \] Input:

int((x*atan(a*x)^(3/2))/(c + a^2*c*x^2)^(3/2),x)
 

Output:

int((x*atan(a*x)^(3/2))/(c + a^2*c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-4 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )+6 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, a x -3 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{4} x^{4}+2 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{4} x^{2}-3 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{4} x^{4}+2 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{2}\right )}{4 a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \] Input:

int(x*atan(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)
 

Output:

(sqrt(c)*( - 4*sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*atan(a*x) + 6*sqrt(a**2 
*x**2 + 1)*sqrt(atan(a*x))*a*x - 3*int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x) 
)*x)/(atan(a*x)*a**4*x**4 + 2*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**4*x** 
2 - 3*int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*x)/(atan(a*x)*a**4*x**4 + 2 
*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**2))/(4*a**2*c**2*(a**2*x**2 + 1))