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

Optimal. Leaf size=161 \[ -\frac{c^3 \left (a^2 x^2+1\right )^3}{42 a}-\frac{3 c^3 \left (a^2 x^2+1\right )^2}{70 a}-\frac{4 c^3 \left (a^2 x^2+1\right )}{35 a}-\frac{8 c^3 \log \left (a^2 x^2+1\right )}{35 a}+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x) \]

[Out]

(-4*c^3*(1 + a^2*x^2))/(35*a) - (3*c^3*(1 + a^2*x^2)^2)/(70*a) - (c^3*(1 + a^2*x^2)^3)/(42*a) + (16*c^3*x*ArcT
an[a*x])/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 + (c^3*x*(1 +
a^2*x^2)^3*ArcTan[a*x])/7 - (8*c^3*Log[1 + a^2*x^2])/(35*a)

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Rubi [A]  time = 0.0763048, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4878, 4846, 260} \[ -\frac{c^3 \left (a^2 x^2+1\right )^3}{42 a}-\frac{3 c^3 \left (a^2 x^2+1\right )^2}{70 a}-\frac{4 c^3 \left (a^2 x^2+1\right )}{35 a}-\frac{8 c^3 \log \left (a^2 x^2+1\right )}{35 a}+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)^3*ArcTan[a*x],x]

[Out]

(-4*c^3*(1 + a^2*x^2))/(35*a) - (3*c^3*(1 + a^2*x^2)^2)/(70*a) - (c^3*(1 + a^2*x^2)^3)/(42*a) + (16*c^3*x*ArcT
an[a*x])/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 + (c^3*x*(1 +
a^2*x^2)^3*ArcTan[a*x])/7 - (8*c^3*Log[1 + a^2*x^2])/(35*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 )^3 \tan ^{-1}(a x) \, dx &=-\frac{c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)+\frac{1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx\\ &=-\frac{3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)+\frac{1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx\\ &=-\frac{4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)+\frac{1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac{4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)-\frac{1}{35} \left (16 a c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx\\ &=-\frac{4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)-\frac{8 c^3 \log \left (1+a^2 x^2\right )}{35 a}\\ \end{align*}

Mathematica [A]  time = 0.076559, size = 83, normalized size = 0.52 \[ \frac{c^3 \left (-a^2 x^2 \left (5 a^4 x^4+24 a^2 x^2+57\right )-48 \log \left (a^2 x^2+1\right )+6 a x \left (5 a^6 x^6+21 a^4 x^4+35 a^2 x^2+35\right ) \tan ^{-1}(a x)\right )}{210 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-(a^2*x^2*(57 + 24*a^2*x^2 + 5*a^4*x^4)) + 6*a*x*(35 + 35*a^2*x^2 + 21*a^4*x^4 + 5*a^6*x^6)*ArcTan[a*x]
- 48*Log[1 + a^2*x^2]))/(210*a)

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Maple [A]  time = 0.026, size = 104, normalized size = 0.7 \begin{align*}{\frac{{a}^{6}{c}^{3}\arctan \left ( ax \right ){x}^{7}}{7}}+{\frac{3\,{a}^{4}{c}^{3}\arctan \left ( ax \right ){x}^{5}}{5}}+{a}^{2}{c}^{3}\arctan \left ( ax \right ){x}^{3}+{c}^{3}x\arctan \left ( ax \right ) -{\frac{{a}^{5}{c}^{3}{x}^{6}}{42}}-{\frac{4\,{a}^{3}{c}^{3}{x}^{4}}{35}}-{\frac{19\,a{c}^{3}{x}^{2}}{70}}-{\frac{8\,{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{35\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^6*c^3*arctan(a*x)*x^7+3/5*a^4*c^3*arctan(a*x)*x^5+a^2*c^3*arctan(a*x)*x^3+c^3*x*arctan(a*x)-1/42*a^5*c^3
*x^6-4/35*a^3*c^3*x^4-19/70*a*c^3*x^2-8/35*c^3*ln(a^2*x^2+1)/a

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Maxima [A]  time = 0.966886, size = 134, normalized size = 0.83 \begin{align*} -\frac{1}{210} \,{\left (5 \, a^{4} c^{3} x^{6} + 24 \, a^{2} c^{3} x^{4} + 57 \, c^{3} x^{2} + \frac{48 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac{1}{35} \,{\left (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\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)^3*arctan(a*x),x, algorithm="maxima")

[Out]

-1/210*(5*a^4*c^3*x^6 + 24*a^2*c^3*x^4 + 57*c^3*x^2 + 48*c^3*log(a^2*x^2 + 1)/a^2)*a + 1/35*(5*a^6*c^3*x^7 + 2
1*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*arctan(a*x)

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Fricas [A]  time = 1.65825, size = 223, normalized size = 1.39 \begin{align*} -\frac{5 \, a^{6} c^{3} x^{6} + 24 \, a^{4} c^{3} x^{4} + 57 \, a^{2} c^{3} x^{2} + 48 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 6 \,{\left (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\right )} \arctan \left (a x\right )}{210 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(5*a^6*c^3*x^6 + 24*a^4*c^3*x^4 + 57*a^2*c^3*x^2 + 48*c^3*log(a^2*x^2 + 1) - 6*(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)*arctan(a*x))/a

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Sympy [A]  time = 2.76145, size = 117, normalized size = 0.73 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{7} \operatorname{atan}{\left (a x \right )}}{7} - \frac{a^{5} c^{3} x^{6}}{42} + \frac{3 a^{4} c^{3} x^{5} \operatorname{atan}{\left (a x \right )}}{5} - \frac{4 a^{3} c^{3} x^{4}}{35} + a^{2} c^{3} x^{3} \operatorname{atan}{\left (a x \right )} - \frac{19 a c^{3} x^{2}}{70} + c^{3} x \operatorname{atan}{\left (a x \right )} - \frac{8 c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{35 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)**3*atan(a*x),x)

[Out]

Piecewise((a**6*c**3*x**7*atan(a*x)/7 - a**5*c**3*x**6/42 + 3*a**4*c**3*x**5*atan(a*x)/5 - 4*a**3*c**3*x**4/35
+ a**2*c**3*x**3*atan(a*x) - 19*a*c**3*x**2/70 + c**3*x*atan(a*x) - 8*c**3*log(x**2 + a**(-2))/(35*a), Ne(a,
0)), (0, True))

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Giac [A]  time = 1.13563, size = 140, normalized size = 0.87 \begin{align*} -\frac{8 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{35 \, a} + \frac{1}{35} \,{\left (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\right )} \arctan \left (a x\right ) - \frac{5 \, a^{11} c^{3} x^{6} + 24 \, a^{9} c^{3} x^{4} + 57 \, a^{7} c^{3} x^{2}}{210 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/35*c^3*log(a^2*x^2 + 1)/a + 1/35*(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)*arctan(a*x) -
1/210*(5*a^11*c^3*x^6 + 24*a^9*c^3*x^4 + 57*a^7*c^3*x^2)/a^6