\(\int \frac {x^3}{(c+a^2 c x^2)^{5/2} \arctan (a x)^{5/2}} \, dx\) [1102]

Optimal result

Integrand size = 26, antiderivative size = 190 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=-\frac {2 x^3}{3 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}-\frac {4 x^2}{a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {6 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{a^4 c^2 \sqrt {c+a^2 c x^2}} \] Output:

-2/3*x^3/a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2)-4*x^2/a^2/c/(a^2*c*x^2+ 
c)^(3/2)/arctan(a*x)^(1/2)-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^4/c^2/(a^2*c*x^2+c)^(1/2)+6^(1/2)*Pi^(1 
/2)*(a^2*x^2+1)^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a^4/c^2 
/(a^2*c*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.34 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\frac {-2 a^2 x^2 (a x+6 \arctan (a x))+\sqrt {6 \pi } \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^{3/2} \left (-3 \sqrt {3} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )-\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \left (3 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+3 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )+\sqrt {3} \left (\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )\right )\right )}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \] Input:

Integrate[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]
 

Output:

(-2*a^2*x^2*(a*x + 6*ArcTan[a*x]) + Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*ArcTan[ 
a*x]^(3/2)*(-3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + FresnelS[S 
qrt[6/Pi]*Sqrt[ArcTan[a*x]]]) - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*(3*Sqrt[(- 
I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma 
[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*A 
rcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(3*a^4* 
c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.45, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5477, 5503, 5506, 5505, 3042, 3793, 2009, 4906, 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.

\(\displaystyle \int \frac {x^3}{\arctan (a x)^{5/2} \left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5477

\(\displaystyle \frac {2 \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a c \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5503

\(\displaystyle \frac {2 \left (\frac {4 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-2 a \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {2 x^2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{a}-\frac {2 x^3}{3 a c \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5506

\(\displaystyle \frac {2 \left (\frac {4 \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {2 a \sqrt {a^2 x^2+1} \int \frac {x^3}{\left (a^2 x^2+1\right )^{5/2} \sqrt {\arctan (a x)}}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x^2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{a}-\frac {2 x^3}{3 a c \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {4 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}d\arctan (a x)}{a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x^2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a^3 c^2 \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {2 x^3}{3 a c \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {2 \left (\frac {4 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}+\frac {\sin (3 \arctan (a x))}{4 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x^2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a^3 c^2 \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {2 x^3}{3 a c \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {2 x^2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {4 \sqrt {a^2 x^2+1} \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a^3 c^2 \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {2 x^3}{3 a c \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\)

Input:

Int[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]
 

Output:

(-2*x^3)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + (2*((-2*x^2)/(a 
*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (2*Sqrt[1 + a^2*x^2]*((3*Sqr 
t[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/2 - (Sqrt[Pi/6]*FresnelS[S 
qrt[6/Pi]*Sqrt[ArcTan[a*x]]])/2))/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (4*Sqrt[ 
1 + a^2*x^2]*((Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/2 + (Sqr 
t[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/2))/(a^3*c^2*Sqrt[c + a^2* 
c*x^2])))/a
 

Defintions of rubi rules used

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 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 5477

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_. 
)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcT 
an[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[f*(m/(b*c*(p + 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 + 2, 0] && LtQ[p, -1 
]
 

rule 5503

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

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])
 

rule 5506

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) 
  Int[x^m*(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] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(I 
ntegerQ[q] || GtQ[d, 0])
 

Maple [F]

\[\int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]

Input:

int(x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x)
 

Output:

int(x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**3/(a**2*c*x**2+c)**(5/2)/atan(a*x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:

int(x^3/(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(5/2)),x)
 

Output:

int(x^3/(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(5/2)), x)
 

Reduce [F]

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

int(x^3/(a^2*c*x^2+c)^(5/2)/atan(a*x)^(5/2),x)
 

Output:

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