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

Optimal. Leaf size=50 \[ -\frac{c \log \left (a^2 x^2+1\right )}{3 a}+\frac{1}{3} a^2 c x^3 \tan ^{-1}(a x)-\frac{1}{6} a c x^2+c x \tan ^{-1}(a x) \]

[Out]

-(a*c*x^2)/6 + c*x*ArcTan[a*x] + (a^2*c*x^3*ArcTan[a*x])/3 - (c*Log[1 + a^2*x^2])/(3*a)

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Rubi [A]  time = 0.0234937, antiderivative size = 65, normalized size of antiderivative = 1.3, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4878, 4846, 260} \[ -\frac{c \left (a^2 x^2+1\right )}{6 a}-\frac{c \log \left (a^2 x^2+1\right )}{3 a}+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac{2}{3} c x \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)*ArcTan[a*x],x]

[Out]

-(c*(1 + a^2*x^2))/(6*a) + (2*c*x*ArcTan[a*x])/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x])/3 - (c*Log[1 + a^2*x^2])/(3
*a)

Rule 4878

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> -Simp[(b*(d + e*x^2)^q)/(2*c*
q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x] + Simp[(x*(d +
e*x^2)^q*(a + b*ArcTan[c*x]))/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]

Rule 4846

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

Rule 260

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]

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx &=-\frac{c \left (1+a^2 x^2\right )}{6 a}+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{3} (2 c) \int \tan ^{-1}(a x) \, dx\\ &=-\frac{c \left (1+a^2 x^2\right )}{6 a}+\frac{2}{3} c x \tan ^{-1}(a x)+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac{1}{3} (2 a c) \int \frac{x}{1+a^2 x^2} \, dx\\ &=-\frac{c \left (1+a^2 x^2\right )}{6 a}+\frac{2}{3} c x \tan ^{-1}(a x)+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac{c \log \left (1+a^2 x^2\right )}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0103881, size = 50, normalized size = 1. \[ -\frac{c \log \left (a^2 x^2+1\right )}{3 a}+\frac{1}{3} a^2 c x^3 \tan ^{-1}(a x)-\frac{1}{6} a c x^2+c x \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*c*x^2)/6 + c*x*ArcTan[a*x] + (a^2*c*x^3*ArcTan[a*x])/3 - (c*Log[1 + a^2*x^2])/(3*a)

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Maple [A]  time = 0.024, size = 45, normalized size = 0.9 \begin{align*} -{\frac{a{x}^{2}c}{6}}+cx\arctan \left ( ax \right ) +{\frac{{a}^{2}c{x}^{3}\arctan \left ( ax \right ) }{3}}-{\frac{c\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975009, size = 61, normalized size = 1.22 \begin{align*} -\frac{1}{6} \,{\left (c x^{2} + \frac{2 \, c \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac{1}{3} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \arctan \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.61202, size = 109, normalized size = 2.18 \begin{align*} -\frac{a^{2} c x^{2} - 2 \,{\left (a^{3} c x^{3} + 3 \, a c x\right )} \arctan \left (a x\right ) + 2 \, c \log \left (a^{2} x^{2} + 1\right )}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.747634, size = 48, normalized size = 0.96 \begin{align*} \begin{cases} \frac{a^{2} c x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{a c x^{2}}{6} + c x \operatorname{atan}{\left (a x \right )} - \frac{c \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{3 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c*x**3*atan(a*x)/3 - a*c*x**2/6 + c*x*atan(a*x) - c*log(x**2 + a**(-2))/(3*a), Ne(a, 0)), (0,
True))

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Giac [A]  time = 1.12351, size = 58, normalized size = 1.16 \begin{align*} -\frac{1}{6} \, a c x^{2} + \frac{1}{3} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \arctan \left (a x\right ) - \frac{c \log \left (a^{2} x^{2} + 1\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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