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

Optimal. Leaf size=74 \[ -\frac{1}{56} a^5 c^3 x^7-\frac{3}{40} a^3 c^3 x^5+\frac{c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac{1}{8} a c^3 x^3-\frac{c^3 x}{8 a} \]

[Out]

-(c^3*x)/(8*a) - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x])/(8*
a^2)

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Rubi [A]  time = 0.050066, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4930, 194} \[ -\frac{1}{56} a^5 c^3 x^7-\frac{3}{40} a^3 c^3 x^5+\frac{c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac{1}{8} a c^3 x^3-\frac{c^3 x}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*x)/(8*a) - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x])/(8*
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]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac{\int \left (c+a^2 c x^2\right )^3 \, dx}{8 a}\\ &=\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac{\int \left (c^3+3 a^2 c^3 x^2+3 a^4 c^3 x^4+a^6 c^3 x^6\right ) \, dx}{8 a}\\ &=-\frac{c^3 x}{8 a}-\frac{1}{8} a c^3 x^3-\frac{3}{40} a^3 c^3 x^5-\frac{1}{56} a^5 c^3 x^7+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)}{8 a^2}\\ \end{align*}

Mathematica [A]  time = 0.087915, size = 128, normalized size = 1.73 \[ -\frac{1}{56} a^5 c^3 x^7-\frac{3}{40} a^3 c^3 x^5+\frac{1}{8} a^6 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{2} a^4 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 x^4 \tan ^{-1}(a x)+\frac{c^3 \tan ^{-1}(a x)}{8 a^2}-\frac{1}{8} a c^3 x^3+\frac{1}{2} c^3 x^2 \tan ^{-1}(a x)-\frac{c^3 x}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*x)/(8*a) - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*ArcTan[a*x])/(8*a^2) + (c^3*x^2*
ArcTan[a*x])/2 + (3*a^2*c^3*x^4*ArcTan[a*x])/4 + (a^4*c^3*x^6*ArcTan[a*x])/2 + (a^6*c^3*x^8*ArcTan[a*x])/8

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Maple [A]  time = 0.025, size = 111, normalized size = 1.5 \begin{align*}{\frac{{a}^{6}{c}^{3}\arctan \left ( ax \right ){x}^{8}}{8}}+{\frac{{a}^{4}{c}^{3}\arctan \left ( ax \right ){x}^{6}}{2}}+{\frac{3\,{a}^{2}{c}^{3}\arctan \left ( ax \right ){x}^{4}}{4}}+{\frac{{c}^{3}\arctan \left ( ax \right ){x}^{2}}{2}}-{\frac{{a}^{5}{c}^{3}{x}^{7}}{56}}-{\frac{3\,{a}^{3}{c}^{3}{x}^{5}}{40}}-{\frac{{c}^{3}{x}^{3}a}{8}}-{\frac{{c}^{3}x}{8\,a}}+{\frac{{c}^{3}\arctan \left ( ax \right ) }{8\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^6*c^3*arctan(a*x)*x^8+1/2*a^4*c^3*arctan(a*x)*x^6+3/4*a^2*c^3*arctan(a*x)*x^4+1/2*c^3*arctan(a*x)*x^2-1/
56*a^5*c^3*x^7-3/40*a^3*c^3*x^5-1/8*c^3*x^3*a-1/8*c^3*x/a+1/8/a^2*c^3*arctan(a*x)

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Maxima [A]  time = 0.972201, size = 99, normalized size = 1.34 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )}{8 \, a^{2} c} - \frac{5 \, a^{6} c^{4} x^{7} + 21 \, a^{4} c^{4} x^{5} + 35 \, a^{2} c^{4} x^{3} + 35 \, c^{4} x}{280 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(a^2*c*x^2 + c)^4*arctan(a*x)/(a^2*c) - 1/280*(5*a^6*c^4*x^7 + 21*a^4*c^4*x^5 + 35*a^2*c^4*x^3 + 35*c^4*x)
/(a*c)

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Fricas [A]  time = 1.70776, size = 216, normalized size = 2.92 \begin{align*} -\frac{5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x - 35 \,{\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}{280 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/280*(5*a^7*c^3*x^7 + 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x - 35*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4
*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x))/a^2

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Sympy [A]  time = 3.50299, size = 124, normalized size = 1.68 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{8} \operatorname{atan}{\left (a x \right )}}{8} - \frac{a^{5} c^{3} x^{7}}{56} + \frac{a^{4} c^{3} x^{6} \operatorname{atan}{\left (a x \right )}}{2} - \frac{3 a^{3} c^{3} x^{5}}{40} + \frac{3 a^{2} c^{3} x^{4} \operatorname{atan}{\left (a x \right )}}{4} - \frac{a c^{3} x^{3}}{8} + \frac{c^{3} x^{2} \operatorname{atan}{\left (a x \right )}}{2} - \frac{c^{3} x}{8 a} + \frac{c^{3} \operatorname{atan}{\left (a x \right )}}{8 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)**3*atan(a*x),x)

[Out]

Piecewise((a**6*c**3*x**8*atan(a*x)/8 - a**5*c**3*x**7/56 + a**4*c**3*x**6*atan(a*x)/2 - 3*a**3*c**3*x**5/40 +
3*a**2*c**3*x**4*atan(a*x)/4 - a*c**3*x**3/8 + c**3*x**2*atan(a*x)/2 - c**3*x/(8*a) + c**3*atan(a*x)/(8*a**2)
, Ne(a, 0)), (0, True))

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Giac [A]  time = 1.12103, size = 95, normalized size = 1.28 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )}{8 \, a^{2} c} - \frac{5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x}{280 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(a^2*c*x^2 + c)^4*arctan(a*x)/(a^2*c) - 1/280*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)
/a