\(\int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx\) [175]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {x \arctan (a x)}{a^2 c}-\frac {\arctan (a x)^2}{2 a^3 c}-\frac {\log \left (1+a^2 x^2\right )}{2 a^3 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5036, 4930, 266, 5004} \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {\arctan (a x)^2}{2 a^3 c}+\frac {x \arctan (a x)}{a^2 c}-\frac {\log \left (a^2 x^2+1\right )}{2 a^3 c} \]

[In]

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

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rule 5004

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

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {x \arctan (a x)}{a^2 c}-\frac {\arctan (a x)^2}{2 a^3 c}-\frac {\log \left (1+a^2 x^2\right )}{2 a^3 c} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {2 x \arctan \left (a x \right ) a -\arctan \left (a x \right )^{2}-\ln \left (a^{2} x^{2}+1\right )}{2 c \,a^{3}}\)	\(38\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right ) a x}{c}-\frac {\arctan \left (a x \right )^{2}}{c}-\frac {\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{c}}{a^{3}}\)	\(53\)
default	\(\frac {\frac {\arctan \left (a x \right ) a x}{c}-\frac {\arctan \left (a x \right )^{2}}{c}-\frac {\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{c}}{a^{3}}\)	\(53\)
parts	\(\frac {x \arctan \left (a x \right )}{a^{2} c}-\frac {\arctan \left (a x \right )^{2}}{a^{3} c}-\frac {\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a^{3}}-\frac {\arctan \left (a x \right )^{2}}{2 a^{3}}}{c}\)	\(60\)
risch	\(\frac {\ln \left (i a x +1\right )^{2}}{8 a^{3} c}-\frac {i \left (-i \ln \left (-i a x +1\right )+2 a x \right ) \ln \left (i a x +1\right )}{4 c \,a^{3}}+\frac {\ln \left (-i a x +1\right )^{2}}{8 c \,a^{3}}+\frac {i x \ln \left (-i a x +1\right )}{2 c \,a^{2}}-\frac {\ln \left (-a^{2} x^{2}-1\right )}{2 c \,a^{3}}\)	\(108\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {2 \, a x \arctan \left (a x\right ) - \arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{3} c} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx=\begin {cases} \frac {x \operatorname {atan}{\left (a x \right )}}{a^{2} c} - \frac {\log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a^{3} c} - \frac {\operatorname {atan}^{2}{\left (a x \right )}}{2 a^{3} c} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x*atan(a*x)/(a**2*c) - log(x**2 + a**(-2))/(2*a**3*c) - atan(a*x)**2/(2*a**3*c), Ne(a, 0)), (0, Tru
e))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx={\left (\frac {x}{a^{2} c} - \frac {\arctan \left (a x\right )}{a^{3} c}\right )} \arctan \left (a x\right ) + \frac {\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{3} c} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {{\mathrm {atan}\left (a\,x\right )}^2-2\,a\,x\,\mathrm {atan}\left (a\,x\right )+\ln \left (a^2\,x^2+1\right )}{2\,a^3\,c} \]

[In]

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

[Out]

-(log(a^2*x^2 + 1) + atan(a*x)^2 - 2*a*x*atan(a*x))/(2*a^3*c)