3.149 \(\int x^3 (c+a^2 c x^2) \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac{c x}{12 a^3}-\frac{c \tan ^{-1}(a x)}{12 a^4}-\frac{1}{30} a c x^5-\frac{c x^3}{36 a}+\frac{1}{4} c x^4 \tan ^{-1}(a x) \]

[Out]

(c*x)/(12*a^3) - (c*x^3)/(36*a) - (a*c*x^5)/30 - (c*ArcTan[a*x])/(12*a^4) + (c*x^4*ArcTan[a*x])/4 + (a^2*c*x^6
*ArcTan[a*x])/6

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Rubi [A]  time = 0.0855232, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4950, 4852, 302, 203} \[ \frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac{c x}{12 a^3}-\frac{c \tan ^{-1}(a x)}{12 a^4}-\frac{1}{30} a c x^5-\frac{c x^3}{36 a}+\frac{1}{4} c x^4 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*(c + a^2*c*x^2)*ArcTan[a*x],x]

[Out]

(c*x)/(12*a^3) - (c*x^3)/(36*a) - (a*c*x^5)/30 - (c*ArcTan[a*x])/(12*a^4) + (c*x^4*ArcTan[a*x])/4 + (a^2*c*x^6
*ArcTan[a*x])/6

Rule 4950

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

Rule 4852

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx &=c \int x^3 \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^5 \tan ^{-1}(a x) \, dx\\ &=\frac{1}{4} c x^4 \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac{1}{4} (a c) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{6} \left (a^3 c\right ) \int \frac{x^6}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c x^4 \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac{1}{4} (a c) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{6} \left (a^3 c\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{c x}{12 a^3}-\frac{c x^3}{36 a}-\frac{1}{30} a c x^5+\frac{1}{4} c x^4 \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{6 a^3}-\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}\\ &=\frac{c x}{12 a^3}-\frac{c x^3}{36 a}-\frac{1}{30} a c x^5-\frac{c \tan ^{-1}(a x)}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0054891, size = 69, normalized size = 1. \[ \frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac{c x}{12 a^3}-\frac{c \tan ^{-1}(a x)}{12 a^4}-\frac{1}{30} a c x^5-\frac{c x^3}{36 a}+\frac{1}{4} c x^4 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(c + a^2*c*x^2)*ArcTan[a*x],x]

[Out]

(c*x)/(12*a^3) - (c*x^3)/(36*a) - (a*c*x^5)/30 - (c*ArcTan[a*x])/(12*a^4) + (c*x^4*ArcTan[a*x])/4 + (a^2*c*x^6
*ArcTan[a*x])/6

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Maple [A]  time = 0.023, size = 58, normalized size = 0.8 \begin{align*}{\frac{cx}{12\,{a}^{3}}}-{\frac{c{x}^{3}}{36\,a}}-{\frac{ac{x}^{5}}{30}}-{\frac{c\arctan \left ( ax \right ) }{12\,{a}^{4}}}+{\frac{c{x}^{4}\arctan \left ( ax \right ) }{4}}+{\frac{{a}^{2}c{x}^{6}\arctan \left ( ax \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a^2*c*x^2+c)*arctan(a*x),x)

[Out]

1/12*c*x/a^3-1/36*c*x^3/a-1/30*a*c*x^5-1/12*c*arctan(a*x)/a^4+1/4*c*x^4*arctan(a*x)+1/6*a^2*c*x^6*arctan(a*x)

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Maxima [A]  time = 1.48318, size = 86, normalized size = 1.25 \begin{align*} -\frac{1}{180} \, a{\left (\frac{6 \, a^{4} c x^{5} + 5 \, a^{2} c x^{3} - 15 \, c x}{a^{4}} + \frac{15 \, c \arctan \left (a x\right )}{a^{5}}\right )} + \frac{1}{12} \,{\left (2 \, a^{2} c x^{6} + 3 \, c x^{4}\right )} \arctan \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)*arctan(a*x),x, algorithm="maxima")

[Out]

-1/180*a*((6*a^4*c*x^5 + 5*a^2*c*x^3 - 15*c*x)/a^4 + 15*c*arctan(a*x)/a^5) + 1/12*(2*a^2*c*x^6 + 3*c*x^4)*arct
an(a*x)

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Fricas [A]  time = 1.67906, size = 135, normalized size = 1.96 \begin{align*} -\frac{6 \, a^{5} c x^{5} + 5 \, a^{3} c x^{3} - 15 \, a c x - 15 \,{\left (2 \, a^{6} c x^{6} + 3 \, a^{4} c x^{4} - c\right )} \arctan \left (a x\right )}{180 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)*arctan(a*x),x, algorithm="fricas")

[Out]

-1/180*(6*a^5*c*x^5 + 5*a^3*c*x^3 - 15*a*c*x - 15*(2*a^6*c*x^6 + 3*a^4*c*x^4 - c)*arctan(a*x))/a^4

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Sympy [A]  time = 1.85988, size = 65, normalized size = 0.94 \begin{align*} \begin{cases} \frac{a^{2} c x^{6} \operatorname{atan}{\left (a x \right )}}{6} - \frac{a c x^{5}}{30} + \frac{c x^{4} \operatorname{atan}{\left (a x \right )}}{4} - \frac{c x^{3}}{36 a} + \frac{c x}{12 a^{3}} - \frac{c \operatorname{atan}{\left (a x \right )}}{12 a^{4}} & \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**3*(a**2*c*x**2+c)*atan(a*x),x)

[Out]

Piecewise((a**2*c*x**6*atan(a*x)/6 - a*c*x**5/30 + c*x**4*atan(a*x)/4 - c*x**3/(36*a) + c*x/(12*a**3) - c*atan
(a*x)/(12*a**4), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.12661, size = 86, normalized size = 1.25 \begin{align*} \frac{1}{12} \,{\left (2 \, a^{2} c x^{6} + 3 \, c x^{4}\right )} \arctan \left (a x\right ) - \frac{c \arctan \left (a x\right )}{12 \, a^{4}} - \frac{6 \, a^{11} c x^{5} + 5 \, a^{9} c x^{3} - 15 \, a^{7} c x}{180 \, a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)*arctan(a*x),x, algorithm="giac")

[Out]

1/12*(2*a^2*c*x^6 + 3*c*x^4)*arctan(a*x) - 1/12*c*arctan(a*x)/a^4 - 1/180*(6*a^11*c*x^5 + 5*a^9*c*x^3 - 15*a^7
*c*x)/a^10