\(\int \frac {x^3 \arctan (a x)^{3/2}}{(c+a^2 c x^2)^3} \, dx\) [791]

Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \arctan (a x)^{3/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^4 c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{64 a^4 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^4 c^3}-\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^4 c^3} \] Output:

-3/32*arctan(a*x)^(3/2)/a^4/c^3+1/4*x^4*arctan(a*x)^(3/2)/c^3/(a^2*x^2+1)^ 
2+3/1024*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a 
^4/c^3-3/64*Pi^(1/2)*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))/a^4/c^3+3/32*a 
rctan(a*x)^(1/2)*sin(2*arctan(a*x))/a^4/c^3-3/256*arctan(a*x)^(1/2)*sin(4* 
arctan(a*x))/a^4/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.08 \[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\sqrt {\arctan (a x)} \left (\frac {3 x \left (3+5 a^2 x^2\right )}{64 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\left (-3-6 a^2 x^2+5 a^4 x^4\right ) \arctan (a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )^2}\right )-\frac {9 \left (-2 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-2 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{4096 a^4 c^3 \sqrt {\arctan (a x)}}-\frac {15 \left (-2 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-2 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{4096 a^4 c^3 \sqrt {\arctan (a x)}} \] Input:

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

Output:

Sqrt[ArcTan[a*x]]*((3*x*(3 + 5*a^2*x^2))/(64*a^3*c^3*(1 + a^2*x^2)^2) + (( 
-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)^2)) - ( 
9*(-2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sq 
rt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - Sqrt[(-I)*ArcTan 
[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4* 
I)*ArcTan[a*x]]))/(4096*a^4*c^3*Sqrt[ArcTan[a*x]]) - (15*(-2*Sqrt[2]*Sqrt[ 
(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcTan 
[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, ( 
-4*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/( 
4096*a^4*c^3*Sqrt[ArcTan[a*x]])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5479, 27, 5505, 3042, 3793, 2009}

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.

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^4 \arctan (a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \left (-\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{8} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {1}{4} \arctan (a x)^{3/2}-\frac {1}{4} \sqrt {\arctan (a x)} \sin (2 \arctan (a x))+\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))\right )}{8 a^4 c^3}\)

Input:

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

Output:

(x^4*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) - (3*(ArcTan[a*x]^(3/2)/4 
- (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/64 + (Sqrt[Pi]*Fre 
snelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/8 - (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan 
[a*x]])/4 + (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/32))/(8*a^4*c^3)
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -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 = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.74

method	result	size
default	\(\frac {3 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+32 \arctan \left (a x \right )^{2} \cos \left (4 \arctan \left (a x \right )\right )-128 \arctan \left (a x \right )^{2} \cos \left (2 \arctan \left (a x \right )\right )-48 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-12 \arctan \left (a x \right ) \sin \left (4 \arctan \left (a x \right )\right )+96 \arctan \left (a x \right ) \sin \left (2 \arctan \left (a x \right )\right )}{1024 a^{4} c^{3} \sqrt {\arctan \left (a x \right )}}\)	\(124\)

Input:

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

Output:

1/1024/a^4/c^3*(3*2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi 
^(1/2)*arctan(a*x)^(1/2))+32*arctan(a*x)^2*cos(4*arctan(a*x))-128*arctan(a 
*x)^2*cos(2*arctan(a*x))-48*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arctan(a 
*x)^(1/2)/Pi^(1/2))-12*arctan(a*x)*sin(4*arctan(a*x))+96*arctan(a*x)*sin(2 
*arctan(a*x)))/arctan(a*x)^(1/2)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x**3*atan(a*x)**(3/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1) 
, x)/c**3
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-4 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}-2 \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )-2 \sqrt {\mathit {atan} \left (a x \right )}\, a x +5 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{5} x^{4}+10 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{3} x^{2}+5 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a +\left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{6} x^{4}+2 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{4} x^{2}+\left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{2}}{8 a^{4} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 4*sqrt(atan(a*x))*atan(a*x)*a**2*x**2 - 2*sqrt(atan(a*x))*atan(a*x) - 
2*sqrt(atan(a*x))*a*x + 5*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3 
*a**2*x**2 + 1),x)*a**5*x**4 + 10*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4* 
x**4 + 3*a**2*x**2 + 1),x)*a**3*x**2 + 5*int(sqrt(atan(a*x))/(a**6*x**6 + 
3*a**4*x**4 + 3*a**2*x**2 + 1),x)*a + int((sqrt(atan(a*x))*x)/(atan(a*x)*a 
**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a 
**6*x**4 + 2*int((sqrt(atan(a*x))*x)/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a* 
*4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**4*x**2 + int((sqrt(atan 
(a*x))*x)/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2* 
x**2 + atan(a*x)),x)*a**2)/(8*a**4*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))