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]

-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

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Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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]

-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

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fricas [A]  time = 0.61, size = 44, normalized size = 1.05 \[ -\frac {a^{3} c x^{3} + 3 \, a c x - 3 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{12 \, a^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.02, size = 49, normalized size = 1.17 \[ \frac {a^{2} c \arctan \left (a x \right ) x^{4}}{4}+\frac {c \arctan \left (a x \right ) x^{2}}{2}-\frac {a c \,x^{3}}{12}-\frac {c x}{4 a}+\frac {c \arctan \left (a x \right )}{4 a^{2}} \]

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 = 0.33, size = 50, normalized size = 1.19 \[ \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{4 \, a^{2} c} - \frac {a^{2} c^{2} x^{3} + 3 \, c^{2} x}{12 \, a c} \]

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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mupad [B]  time = 0.48, size = 48, normalized size = 1.14 \[ \frac {\frac {c\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {a\,c\,x}{4}}{a^2}+\frac {c\,x^2\,\mathrm {atan}\left (a\,x\right )}{2}-\frac {a\,c\,x^3}{12}+\frac {a^2\,c\,x^4\,\mathrm {atan}\left (a\,x\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.80, size = 54, normalized size = 1.29 \[ \begin {cases} \frac {a^{2} c x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {a c x^{3}}{12} + \frac {c x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {c x}{4 a} + \frac {c \operatorname {atan}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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