\(\int \frac {x \arctan (a x)^{3/2}}{(c+a^2 c x^2)^2} \, dx\) [785]

Optimal result

Integrand size = 22, antiderivative size = 109 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3 x \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a^2 c^2} \] Output:

3/8*x*arctan(a*x)^(1/2)/a/c^2/(a^2*x^2+1)+1/4*arctan(a*x)^(3/2)/a^2/c^2-1/ 
2*arctan(a*x)^(3/2)/a^2/c^2/(a^2*x^2+1)-3/32*Pi^(1/2)*FresnelS(2*arctan(a* 
x)^(1/2)/Pi^(1/2))/a^2/c^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\frac {4 \sqrt {\arctan (a x)} \left (3 a x+2 \left (-1+a^2 x^2\right ) \arctan (a x)\right )}{1+a^2 x^2}-3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a^2 c^2} \] Input:

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

Output:

((4*Sqrt[ArcTan[a*x]]*(3*a*x + 2*(-1 + a^2*x^2)*ArcTan[a*x]))/(1 + a^2*x^2 
) - 3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^2*c^2)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5465, 27, 5427, 5505, 4906, 27, 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.

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

\(\Big \downarrow \) 5465

\(\displaystyle \frac {3 \int \frac {\sqrt {\arctan (a x)}}{c^2 \left (a^2 x^2+1\right )^2}dx}{4 a}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {3 \left (-\frac {1}{4} a \int \frac {x}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 5505

\(\displaystyle \frac {3 \left (-\frac {\int \frac {a x}{\left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {3 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{8 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{8 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {3 \left (-\frac {\int \sin (2 \arctan (a x))d\sqrt {\arctan (a x)}}{4 a}+\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {3 \left (\frac {x \sqrt {\arctan (a x)}}{2 \left (a^2 x^2+1\right )}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arctan (a x)^{3/2}}{3 a}\right )}{4 a c^2}-\frac {\arctan (a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}\)

Input:

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

Output:

-1/2*ArcTan[a*x]^(3/2)/(a^2*c^2*(1 + a^2*x^2)) + (3*((x*Sqrt[ArcTan[a*x]]) 
/(2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(3*a) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ 
ArcTan[a*x]])/Sqrt[Pi]])/(8*a)))/(4*a*c^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5427

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

rule 5505

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.61

method	result	size
default	\(-\frac {8 \arctan \left (a x \right )^{2} \cos \left (2 \arctan \left (a x \right )\right )+3 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arctan \left (a x \right ) \sin \left (2 \arctan \left (a x \right )\right )}{32 a^{2} c^{2} \sqrt {\arctan \left (a x \right )}}\)	\(67\)

Input:

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

Output:

-1/32/a^2/c^2*(8*arctan(a*x)^2*cos(2*arctan(a*x))+3*arctan(a*x)^(1/2)*Pi^( 
1/2)*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))-6*arctan(a*x)*sin(2*arctan(a*x 
)))/arctan(a*x)^(1/2)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x*atan(a*x)**(3/2)/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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