\(\int \frac {x \arctan (a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\) [445]

Optimal result

Integrand size = 22, antiderivative size = 107 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {6 x}{a c \sqrt {c+a^2 c x^2}}+\frac {6 \arctan (a x)}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {3 x \arctan (a x)^2}{a c \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^3}{a^2 c \sqrt {c+a^2 c x^2}} \] Output:

-6*x/a/c/(a^2*c*x^2+c)^(1/2)+6*arctan(a*x)/a^2/c/(a^2*c*x^2+c)^(1/2)+3*x*a 
rctan(a*x)^2/a/c/(a^2*c*x^2+c)^(1/2)-arctan(a*x)^3/a^2/c/(a^2*c*x^2+c)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.57 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-6 a x+6 \arctan (a x)+3 a x \arctan (a x)^2-\arctan (a x)^3\right )}{a^2 c^2 \left (1+a^2 x^2\right )} \] Input:

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

Output:

(Sqrt[c + a^2*c*x^2]*(-6*a*x + 6*ArcTan[a*x] + 3*a*x*ArcTan[a*x]^2 - ArcTa 
n[a*x]^3))/(a^2*c^2*(1 + a^2*x^2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5465, 5433, 208}

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.

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5433

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

\(\Big \downarrow \) 208

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 5433

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ 
Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] 
 + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p 
- 1)   Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 3.67 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.25

method	result	size
default	\(-\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a^{2} c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{2 \left (a^{2} x^{2}+1\right ) a^{2} c^{2}}\)	\(134\)
orering	\(-\frac {4 \left (a^{2} x^{2}+1\right ) \left (6 a^{6} x^{6}-a^{4} x^{4}+a^{2} x^{2}-2\right ) \arctan \left (a x \right )^{3}}{a^{4} x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} x^{2}+1\right )^{2} \left (18 a^{4} x^{4}-7 a^{2} x^{2}+4\right ) \left (\frac {\arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {3 x \arctan \left (a x \right )^{2} a}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )}-\frac {3 x^{2} \arctan \left (a x \right )^{3} c \,a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\right )}{x^{2} a^{4}}-\frac {4 \left (a^{2} x^{2}+1\right )^{3} \left (3 a^{2} x^{2}-1\right ) \left (\frac {6 \arctan \left (a x \right )^{2} a}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )}-\frac {9 \arctan \left (a x \right )^{3} c x \,a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 x \arctan \left (a x \right ) a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )^{2}}-\frac {18 x^{2} \arctan \left (a x \right )^{2} a^{3} c}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \left (a^{2} x^{2}+1\right )}-\frac {6 x^{2} \arctan \left (a x \right )^{2} a^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 x^{3} \arctan \left (a x \right )^{3} c^{2} a^{4}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\right )}{a^{4} x}-\frac {\left (a^{2} x^{2}+1\right )^{4} \left (\frac {18 \arctan \left (a x \right ) a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )^{2}}-\frac {81 \arctan \left (a x \right )^{2} a^{3} c x}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \left (a^{2} x^{2}+1\right )}-\frac {24 \arctan \left (a x \right )^{2} a^{3} x}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )^{2}}+\frac {90 \arctan \left (a x \right )^{3} c^{2} x^{2} a^{4}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}-\frac {9 \arctan \left (a x \right )^{3} c \,a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 x \,a^{3}}{\left (a^{2} x^{2}+1\right )^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {54 x^{2} \arctan \left (a x \right ) a^{4} c}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \left (a^{2} x^{2}+1\right )^{2}}-\frac {36 x^{2} \arctan \left (a x \right ) a^{4}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )^{3}}+\frac {135 x^{3} \arctan \left (a x \right )^{2} a^{5} c^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \left (a^{2} x^{2}+1\right )}+\frac {54 x^{3} \arctan \left (a x \right )^{2} a^{5} c}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \left (a^{2} x^{2}+1\right )^{2}}+\frac {24 x^{3} \arctan \left (a x \right )^{2} a^{5}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (a^{2} x^{2}+1\right )^{3}}-\frac {105 x^{4} \arctan \left (a x \right )^{3} c^{3} a^{6}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\right )}{a^{4}}\)	\(820\)

Input:

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

Output:

-1/2*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(1+I*a*x)*(c*(a*x 
-I)*(a*x+I))^(1/2)/(a^2*x^2+1)/a^2/c^2+1/2*(c*(a*x-I)*(a*x+I))^(1/2)*(I*a* 
x-1)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)^2+6*I)/(a^2*x^2+1)/a^2/c 
^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.58 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (3 \, a x \arctan \left (a x\right )^{2} - \arctan \left (a x\right )^{3} - 6 \, a x + 6 \, \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \] Input:

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

Output:

sqrt(a^2*c*x^2 + c)*(3*a*x*arctan(a*x)^2 - arctan(a*x)^3 - 6*a*x + 6*arcta 
n(a*x))/(a^4*c^2*x^2 + a^2*c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\sqrt {c} {\left (\frac {3 \, x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} a c^{2}} - \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{2}} - \frac {6 \, {\left (\frac {x}{\sqrt {a^{2} x^{2} + 1}} - \frac {\arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a}\right )}}{a c^{2}}\right )} \] Input:

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

Output:

sqrt(c)*(3*x*arctan(a*x)^2/(sqrt(a^2*x^2 + 1)*a*c^2) - arctan(a*x)^3/(sqrt 
(a^2*x^2 + 1)*a^2*c^2) - 6*(x/sqrt(a^2*x^2 + 1) - arctan(a*x)/(sqrt(a^2*x^ 
2 + 1)*a))/(a*c^2))
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {3 \, x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} a c} - \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c} a^{2} c} - \frac {6 \, x}{\sqrt {a^{2} c x^{2} + c} a c} + \frac {6 \, \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c} a^{2} c} \] Input:

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

Output:

3*x*arctan(a*x)^2/(sqrt(a^2*c*x^2 + c)*a*c) - arctan(a*x)^3/(sqrt(a^2*c*x^ 
2 + c)*a^2*c) - 6*x/(sqrt(a^2*c*x^2 + c)*a*c) + 6*arctan(a*x)/(sqrt(a^2*c* 
x^2 + c)*a^2*c)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}+3 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} a x +6 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-6 \sqrt {a^{2} x^{2}+1}\, a x -6 a^{2} x^{2}-6\right )}{a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \] Input:

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

Output:

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