\(\int x (c+a^2 c x^2) \arctan (a x) \, dx\) [151]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=-\frac {c x}{4 a}-\frac {1}{12} a c x^3+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)}{4 a^2} \]

[Out]

-1/4*c*x/a-1/12*a*c*x^3+1/4*c*(a^2*x^2+1)^2*arctan(a*x)/a^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5050} \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=\frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)}{4 a^2}-\frac {1}{12} a c x^3-\frac {c x}{4 a} \]

[In]

Int[x*(c + a^2*c*x^2)*ArcTan[a*x],x]

[Out]

-1/4*(c*x)/a - (a*c*x^3)/12 + (c*(1 + a^2*x^2)^2*ArcTan[a*x])/(4*a^2)

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 {c \left (1+a^2 x^2\right )^2 \arctan (a x)}{4 a^2}-\frac {\int \left (c+a^2 c x^2\right ) \, dx}{4 a} \\ & = -\frac {c x}{4 a}-\frac {1}{12} a c x^3+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)}{4 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=-\frac {c x}{4 a}-\frac {1}{12} a c x^3+\frac {c \arctan (a x)}{4 a^2}+\frac {1}{2} c x^2 \arctan (a x)+\frac {1}{4} a^2 c x^4 \arctan (a x) \]

[In]

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

[Out]

-1/4*(c*x)/a - (a*c*x^3)/12 + (c*ArcTan[a*x])/(4*a^2) + (c*x^2*ArcTan[a*x])/2 + (a^2*c*x^4*ArcTan[a*x])/4

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21

method	result	size
parts	\(\frac {c \arctan \left (a x \right ) a^{2} x^{4}}{4}+\frac {c \arctan \left (a x \right ) x^{2}}{2}+\frac {c \arctan \left (a x \right )}{4 a^{2}}-\frac {c \left (\frac {1}{3} a^{2} x^{3}+x \right )}{4 a}\)	\(51\)
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right ) a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )}{2}+\frac {c \arctan \left (a x \right )}{4}-\frac {c \left (\frac {1}{3} a^{3} x^{3}+a x \right )}{4}}{a^{2}}\)	\(54\)
default	\(\frac {\frac {c \arctan \left (a x \right ) a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )}{2}+\frac {c \arctan \left (a x \right )}{4}-\frac {c \left (\frac {1}{3} a^{3} x^{3}+a x \right )}{4}}{a^{2}}\)	\(54\)
parallelrisch	\(\frac {3 c \arctan \left (a x \right ) a^{4} x^{4}-a^{3} c \,x^{3}+6 a^{2} c \,x^{2} \arctan \left (a x \right )-3 a c x +3 c \arctan \left (a x \right )}{12 a^{2}}\)	\(54\)
meijerg	\(\frac {c \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4 a^{2}}+\frac {c \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4 a^{2}}\)	\(83\)
risch	\(-\frac {i c \left (a^{2} x^{2}+1\right )^{2} \ln \left (i a x +1\right )}{8 a^{2}}+\frac {i c \,a^{2} x^{4} \ln \left (-i a x +1\right )}{8}-\frac {a c \,x^{3}}{12}+\frac {i c \,x^{2} \ln \left (-i a x +1\right )}{4}-\frac {c x}{4 a}+\frac {i c \ln \left (a^{2} x^{2}+1\right )}{16 a^{2}}+\frac {c \arctan \left (a x \right )}{8 a^{2}}\)	\(102\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=-\frac {a^{3} c x^{3} + 3 \, a c x - 3 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{12 \, a^{2}} \]

[In]

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

[Out]

-1/12*(a^3*c*x^3 + 3*a*c*x - 3*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x))/a^2

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.29 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=\begin {cases} \frac {a^{2} c x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {a c x^{3}}{12} + \frac {c x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {c x}{4 a} + \frac {c \operatorname {atan}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*c*x**4*atan(a*x)/4 - a*c*x**3/12 + c*x**2*atan(a*x)/2 - c*x/(4*a) + c*atan(a*x)/(4*a**2), Ne(a
, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{4 \, a^{2} c} - \frac {a^{2} c^{2} x^{3} + 3 \, c^{2} x}{12 \, a c} \]

[In]

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

[Out]

1/4*(a^2*c*x^2 + c)^2*arctan(a*x)/(a^2*c) - 1/12*(a^2*c^2*x^3 + 3*c^2*x)/(a*c)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x) \, dx=\frac {\frac {c\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {a\,c\,x}{4}}{a^2}+\frac {c\,x^2\,\mathrm {atan}\left (a\,x\right )}{2}-\frac {a\,c\,x^3}{12}+\frac {a^2\,c\,x^4\,\mathrm {atan}\left (a\,x\right )}{4} \]

[In]

int(x*atan(a*x)*(c + a^2*c*x^2),x)

[Out]

((c*atan(a*x))/4 - (a*c*x)/4)/a^2 + (c*x^2*atan(a*x))/2 - (a*c*x^3)/12 + (a^2*c*x^4*atan(a*x))/4