\(\int \frac {\arctan (a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\) [447]

Optimal result

Integrand size = 21, antiderivative size = 100 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {6}{a c \sqrt {c+a^2 c x^2}}-\frac {6 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}+\frac {3 \arctan (a x)^2}{a c \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^3}{c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5018, 5014} \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-\frac {6 x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}-\frac {6}{a c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 5014

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

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]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.56 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-6-6 a x \arctan (a x)+3 \arctan (a x)^2+a x \arctan (a x)^3\right )}{c^2 \left (a+a^3 x^2\right )} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.66 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a \,c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{2 \left (a^{2} x^{2}+1\right ) a \,c^{2}}\)	\(132\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.58 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (a x \arctan \left (a x\right )^{3} - 6 \, a x \arctan \left (a x\right ) + 3 \, \arctan \left (a x\right )^{2} - 6\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c} c} - \frac {3 \, a {\left (\frac {2 \, x \arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a c} - \frac {\arctan \left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} a^{2} c} + \frac {2}{\sqrt {a^{2} x^{2} + 1} a^{2} c}\right )}}{\sqrt {c}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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