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

Optimal. Leaf size=212 \[ -\frac{3 x^3}{128 a c^3 \left (a^2 x^2+1\right )^2}-\frac{45 x}{256 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}-\frac{27 \tan ^{-1}(a x)}{256 a^4 c^3} \]

[Out]

(-3*x^3)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a^3*c^3*(1 + a^2*x^2)) - (27*ArcTan[a*x])/(256*a^4*c^3) - (
3*x^4*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)) + (3*x^3*ArcTan[a*x]^
2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^3)/(32*a^4*c^3
) + (x^4*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2)

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Rubi [A]  time = 0.293902, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4944, 4940, 4936, 4930, 199, 205, 288} \[ -\frac{3 x^3}{128 a c^3 \left (a^2 x^2+1\right )^2}-\frac{45 x}{256 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}-\frac{27 \tan ^{-1}(a x)}{256 a^4 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x^3)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a^3*c^3*(1 + a^2*x^2)) - (27*ArcTan[a*x])/(256*a^4*c^3) - (
3*x^4*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)) + (3*x^3*ArcTan[a*x]^
2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^3)/(32*a^4*c^3
) + (x^4*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2)

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 4940

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

Rule 4936

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(a + b*ArcTan
[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^
2, x], x] - Simp[(x*(a + b*ArcTan[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c
^2*d] && GtQ[p, 0]

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{4} (3 a) \int \frac{x^4 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{32} (3 a) \int \frac{x^4}{\left (c+a^2 c x^2\right )^3} \, dx-\frac{9 \int \frac{x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=-\frac{3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a^2 c}+\frac{9 \int \frac{x^2}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a c}\\ &=-\frac{3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \int \frac{1}{c+a^2 c x^2} \, dx}{256 a^3 c^2}-\frac{9 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^3 c}\\ &=-\frac{3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{45 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}+\frac{9 \tan ^{-1}(a x)}{256 a^4 c^3}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{9 \int \frac{1}{c+a^2 c x^2} \, dx}{64 a^3 c^2}\\ &=-\frac{3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{45 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{27 \tan ^{-1}(a x)}{256 a^4 c^3}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.242411, size = 105, normalized size = 0.5 \[ \frac{-3 a x \left (17 a^2 x^2+15\right )+8 \left (5 a^4 x^4-6 a^2 x^2-3\right ) \tan ^{-1}(a x)^3+24 a x \left (5 a^2 x^2+3\right ) \tan ^{-1}(a x)^2+\left (-51 a^4 x^4+18 a^2 x^2+45\right ) \tan ^{-1}(a x)}{256 a^4 c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*x*(15 + 17*a^2*x^2) + (45 + 18*a^2*x^2 - 51*a^4*x^4)*ArcTan[a*x] + 24*a*x*(3 + 5*a^2*x^2)*ArcTan[a*x]^2
+ 8*(-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x]^3)/(256*a^4*c^3*(1 + a^2*x^2)^2)

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Maple [A]  time = 0.261, size = 220, normalized size = 1. \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{2\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{15\,{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{32\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{9\,x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{32\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{5\, \left ( \arctan \left ( ax \right ) \right ) ^{3}}{32\,{c}^{3}{a}^{4}}}-{\frac{3\,\arctan \left ( ax \right ) }{32\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{15\,\arctan \left ( ax \right ) }{32\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{51\,{x}^{3}}{256\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{45\,x}{256\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{51\,\arctan \left ( ax \right ) }{256\,{c}^{3}{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a^4/c^3*arctan(a*x)^3/(a^2*x^2+1)^2-1/2/a^4/c^3*arctan(a*x)^3/(a^2*x^2+1)+15/32*x^3*arctan(a*x)^2/a/c^3/(a
^2*x^2+1)^2+9/32/a^3/c^3*x/(a^2*x^2+1)^2*arctan(a*x)^2+5/32*arctan(a*x)^3/a^4/c^3-3/32/a^4/c^3/(a^2*x^2+1)^2*a
rctan(a*x)+15/32*arctan(a*x)/a^4/c^3/(a^2*x^2+1)-51/256*x^3/a/c^3/(a^2*x^2+1)^2-45/256/a^3/c^3/(a^2*x^2+1)^2*x
-51/256*arctan(a*x)/a^4/c^3

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Maxima [A]  time = 1.6801, size = 390, normalized size = 1.84 \begin{align*} \frac{3}{32} \, a{\left (\frac{5 \, a^{2} x^{3} + 3 \, x}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac{5 \, \arctan \left (a x\right )}{a^{5} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac{{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3}}{4 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac{1}{256} \,{\left (\frac{{\left (51 \, a^{3} x^{3} - 40 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 45 \, a x + 51 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{11} c^{3} x^{4} + 2 \, a^{9} c^{3} x^{2} + a^{7} c^{3}} - \frac{24 \,{\left (5 \, a^{2} x^{2} - 5 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*a*((5*a^2*x^3 + 3*x)/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) + 5*arctan(a*x)/(a^5*c^3))*arctan(a*x)^2 - 1
/4*(2*a^2*x^2 + 1)*arctan(a*x)^3/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) - 1/256*((51*a^3*x^3 - 40*(a^4*x^4 +
2*a^2*x^2 + 1)*arctan(a*x)^3 + 45*a*x + 51*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a^2/(a^11*c^3*x^4 + 2*a^9*c^
3*x^2 + a^7*c^3) - 24*(5*a^2*x^2 - 5*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2 + 4)*a*arctan(a*x)/(a^10*c^3*x^4
+ 2*a^8*c^3*x^2 + a^6*c^3))*a

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Fricas [A]  time = 1.8075, size = 271, normalized size = 1.28 \begin{align*} -\frac{51 \, a^{3} x^{3} - 8 \,{\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 3\right )} \arctan \left (a x\right )^{3} - 24 \,{\left (5 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 45 \, a x + 3 \,{\left (17 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 15\right )} \arctan \left (a x\right )}{256 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256*(51*a^3*x^3 - 8*(5*a^4*x^4 - 6*a^2*x^2 - 3)*arctan(a*x)^3 - 24*(5*a^3*x^3 + 3*a*x)*arctan(a*x)^2 + 45*a
*x + 3*(17*a^4*x^4 - 6*a^2*x^2 - 15)*arctan(a*x))/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3} \operatorname{atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*atan(a*x)**3/(a**2*c*x**2+c)**3,x)

[Out]

Integral(x**3*atan(a*x)**3/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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