3.151 \(\int x (c+a^2 c x^2) \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=42 \[ \frac{c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{4 a^2}-\frac{1}{12} a c x^3-\frac{c x}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.0249456, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {4930} \[ \frac{c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{4 a^2}-\frac{1}{12} a c x^3-\frac{c x}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[x*(c + a^2*c*x^2)*ArcTan[a*x],x]

[Out]

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

Rule 4930

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

Mathematica [A]  time = 0.0042802, size = 58, normalized size = 1.38 \[ \frac{1}{4} a^2 c x^4 \tan ^{-1}(a x)+\frac{c \tan ^{-1}(a x)}{4 a^2}-\frac{1}{12} a c x^3+\frac{1}{2} c x^2 \tan ^{-1}(a x)-\frac{c x}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x)/(4*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

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Maple [A]  time = 0.021, size = 49, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}c\arctan \left ( ax \right ){x}^{4}}{4}}+{\frac{c\arctan \left ( ax \right ){x}^{2}}{2}}-{\frac{ac{x}^{3}}{12}}-{\frac{cx}{4\,a}}+{\frac{c\arctan \left ( ax \right ) }{4\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971041, size = 68, normalized size = 1.62 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.55884, size = 107, normalized size = 2.55 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 1.01982, size = 54, normalized size = 1.29 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.12867, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{4} \,{\left (a^{2} c x^{4} + 2 \, c x^{2}\right )} \arctan \left (a x\right ) + \frac{c \arctan \left (a x\right )}{4 \, a^{2}} - \frac{a^{7} c x^{3} + 3 \, a^{5} c x}{12 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a^2*c*x^4 + 2*c*x^2)*arctan(a*x) + 1/4*c*arctan(a*x)/a^2 - 1/12*(a^7*c*x^3 + 3*a^5*c*x)/a^6