\(\int \frac {x \arctan (a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\) [446]

Optimal result

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

[Out]

-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*arctan(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)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5050, 5018, 197} \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\arctan (a x)^3}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {3 x \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}+\frac {6 \arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {6 x}{a c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 5018

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] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(
3/2), x], x] + Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e,
c^2*d] && GtQ[p, 1]

Rule 5050

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan (a x)^3}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a} \\ & = \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}}-\frac {6 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.57 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-6 a x+6 \arctan (a x)+3 a x \arctan (a x)^2-\arctan (a x)^3\right )}{a^2 c^2 \left (1+a^2 x^2\right )} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.63 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.25

method	result	size
default	\(-\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a^{2} c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{2 \left (a^{2} x^{2}+1\right ) a^{2} c^{2}}\)	\(134\)

[In]

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

[Out]

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

none

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.58 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (3 \, a x \arctan \left (a x\right )^{2} - \arctan \left (a x\right )^{3} - 6 \, a x + 6 \, \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.60 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\sqrt {c} {\left (\frac {3 \, x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} a c^{2}} - \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{2}} - \frac {6 \, {\left (\frac {x}{\sqrt {a^{2} x^{2} + 1}} - \frac {\arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a}\right )}}{a c^{2}}\right )} \]

[In]

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

[Out]

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 [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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