3.2.86 \(\int \frac {x \text {ArcTan}(a x)}{(c+a^2 c x^2)^2} \, dx\) [186]

Optimal. Leaf size=62 \[ \frac {x}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\text {ArcTan}(a x)}{4 a^2 c^2}-\frac {\text {ArcTan}(a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5050, 205, 211} \begin {gather*} -\frac {\text {ArcTan}(a x)}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\text {ArcTan}(a x)}{4 a^2 c^2}+\frac {x}{4 a c^2 \left (a^2 x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 39, normalized size = 0.63 \begin {gather*} \frac {a x+\left (-1+a^2 x^2\right ) \text {ArcTan}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 53, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}}{a^{2}}\)	\(53\)
default	\(\frac {-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}}{a^{2}}\)	\(53\)
risch	\(\frac {i \ln \left (i a x +1\right )}{4 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {i \left (2 \ln \left (-i a x +1\right )+\ln \left (a x -i\right ) a^{2} x^{2}+\ln \left (a x -i\right )-\ln \left (-a x -i\right ) a^{2} x^{2}-\ln \left (-a x -i\right )+2 i a x \right )}{8 \left (a x +i\right ) a^{2} c^{2} \left (a x -i\right )}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.45, size = 59, normalized size = 0.95 \begin {gather*} \frac {\frac {x}{a^{2} c x^{2} + c} + \frac {\arctan \left (a x\right )}{a c}}{4 \, a c} - \frac {\arctan \left (a x\right )}{2 \, {\left (a^{2} c x^{2} + c\right )} a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 3.00, size = 40, normalized size = 0.65 \begin {gather*} \frac {a x + {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.44, size = 82, normalized size = 1.32 \begin {gather*} \begin {cases} \frac {a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} + \frac {a x}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [B]
time = 0.17, size = 40, normalized size = 0.65 \begin {gather*} \frac {a\,x-\mathrm {atan}\left (a\,x\right )+a^2\,x^2\,\mathrm {atan}\left (a\,x\right )}{4\,a^2\,c^2\,\left (a^2\,x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________