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

Optimal. Leaf size=208 \[ -\frac{45 x}{256 a c^3 \left (a^2 x^2+1\right )}-\frac{3 x}{128 a c^3 \left (a^2 x^2+1\right )^2}-\frac{\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac{9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac{45 \tan ^{-1}(a x)}{256 a^2 c^3} \]

[Out]

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

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Rubi [A]  time = 0.177407, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4930, 4900, 4892, 199, 205} \[ -\frac{45 x}{256 a c^3 \left (a^2 x^2+1\right )}-\frac{3 x}{128 a c^3 \left (a^2 x^2+1\right )^2}-\frac{\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac{9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac{45 \tan ^{-1}(a x)}{256 a^2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x])/(256*a^2*c^3) + (3*Ar
cTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x]^2)/(16
*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^3)/(32*a^2*c^3) - ArcT
an[a*x]^3/(4*a^2*c^3*(1 + a^2*x^2)^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 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 4892

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0777072, size = 103, normalized size = 0.5 \[ \frac{-3 a x \left (15 a^2 x^2+17\right )+8 \left (3 a^4 x^4+6 a^2 x^2-5\right ) \tan ^{-1}(a x)^3+24 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)^2-3 \left (15 a^4 x^4+6 a^2 x^2-17\right ) \tan ^{-1}(a x)}{256 c^3 \left (a^3 x^2+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.133, size = 191, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{9\,x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{32\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{3}}{32\,{c}^{3}{a}^{2}}}+{\frac{3\,\arctan \left ( ax \right ) }{32\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{9\,\arctan \left ( ax \right ) }{32\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{45\,{x}^{3}a}{256\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{51\,x}{256\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{45\,\arctan \left ( ax \right ) }{256\,{c}^{3}{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*arctan(a*x)^3/a^2/c^3/(a^2*x^2+1)^2+3/16*x*arctan(a*x)^2/a/c^3/(a^2*x^2+1)^2+9/32*x*arctan(a*x)^2/a/c^3/(
a^2*x^2+1)+3/32*arctan(a*x)^3/a^2/c^3+3/32*arctan(a*x)/a^2/c^3/(a^2*x^2+1)^2+9/32*arctan(a*x)/a^2/c^3/(a^2*x^2
+1)-45/256*a/c^3/(a^2*x^2+1)^2*x^3-51/256*x/a/c^3/(a^2*x^2+1)^2-45/256*arctan(a*x)/a^2/c^3

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Maxima [A]  time = 1.69194, size = 367, normalized size = 1.76 \begin{align*} \frac{3 \,{\left (\frac{3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}} + \frac{3 \, \arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2}}{32 \, a c} - \frac{3 \,{\left (\frac{{\left (15 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{7} c^{2} x^{4} + 2 \, a^{5} c^{2} x^{2} + a^{3} c^{2}} - \frac{8 \,{\left (3 \, a^{2} x^{2} - 3 \,{\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^{6} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{2} c^{2}}\right )}}{256 \, a c} - \frac{\arctan \left (a x\right )^{3}}{4 \,{\left (a^{2} c x^{2} + c\right )}^{2} a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*((3*a^2*x^3 + 5*x)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2) + 3*arctan(a*x)/(a*c^2))*arctan(a*x)^2/(a*c) - 3/2
56*((15*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 + 17*a*x + 15*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x
))*a^2/(a^7*c^2*x^4 + 2*a^5*c^2*x^2 + a^3*c^2) - 8*(3*a^2*x^2 - 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2 + 4)
*a*arctan(a*x)/(a^6*c^2*x^4 + 2*a^4*c^2*x^2 + a^2*c^2))/(a*c) - 1/4*arctan(a*x)^3/((a^2*c*x^2 + c)^2*a^2*c)

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Fricas [A]  time = 1.88186, size = 271, normalized size = 1.3 \begin{align*} -\frac{45 \, a^{3} x^{3} - 8 \,{\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 5\right )} \arctan \left (a x\right )^{3} - 24 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{2} + 51 \, a x + 3 \,{\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )}{256 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x \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*atan(a*x)**3/(a**2*c*x**2+c)**3,x)

[Out]

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

[Out]

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