\(\int \frac {x^3 \sqrt {\arctan (a x)}}{(c+a^2 c x^2)^3} \, dx\) [715]

Optimal result

Integrand size = 24, antiderivative size = 118 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \sqrt {\arctan (a x)}}{32 a^4 c^3}+\frac {x^4 \sqrt {\arctan (a x)}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^4 c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{16 a^4 c^3} \] Output:

-3/32*arctan(a*x)^(1/2)/a^4/c^3+1/4*x^4*arctan(a*x)^(1/2)/c^3/(a^2*x^2+1)^ 
2-1/128*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a^ 
4/c^3+1/16*Pi^(1/2)*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2))/a^4/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.95 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-10 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+80 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {\frac {64 \left (-3-6 a^2 x^2+5 a^4 x^4\right ) \arctan (a x)}{\left (1+a^2 x^2\right )^2}-12 i \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+12 i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+3 i \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-3 i \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\sqrt {\arctan (a x)}}}{2048 a^4 c^3} \] Input:

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

Output:

(-10*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 80*Sqrt[Pi]*Fre 
snelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + ((64*(-3 - 6*a^2*x^2 + 5*a^4*x^4)* 
ArcTan[a*x])/(1 + a^2*x^2)^2 - (12*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma 
[1/2, (-2*I)*ArcTan[a*x]] + (12*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, 
(2*I)*ArcTan[a*x]] + (3*I)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan 
[a*x]] - (3*I)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/Sqrt[Arc 
Tan[a*x]])/(2048*a^4*c^3)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.94, 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.

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x^4*Sqrt[ArcTan[a*x]])/(4*c^3*(1 + a^2*x^2)^2) - ((3*Sqrt[ArcTan[a*x]])/4 
 + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/8 - (Sqrt[Pi]*Fre 
snelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/2)/(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 = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {-\sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+4 \arctan \left (a x \right ) \cos \left (4 \arctan \left (a x \right )\right )-16 \arctan \left (a x \right ) \cos \left (2 \arctan \left (a x \right )\right )+8 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{128 a^{4} c^{3} \sqrt {\arctan \left (a x \right )}}\)	\(94\)

Input:

int(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

1/128/a^4/c^3/arctan(a*x)^(1/2)*(-2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*Fresn 
elC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))+4*arctan(a*x)*cos(4*arctan(a*x)) 
-16*arctan(a*x)*cos(2*arctan(a*x))+8*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelC(2 
*arctan(a*x)^(1/2)/Pi^(1/2)))
 

Fricas [F(-2)]

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

integrate(x^3*arctan(a*x)^(1/2)/(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]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3*arctan(a*x)^(1/2)/(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]

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

integrate(x^3*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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