3.183 \(\int \frac{x^4 \tan ^{-1}(a x)}{(c+a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=96 \[ \frac{1}{4 a^5 c^2 \left (a^2 x^2+1\right )}-\frac{\log \left (a^2 x^2+1\right )}{2 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{4 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{a^4 c^2} \]

[Out]

1/(4*a^5*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(a^4*c^2) + (x*ArcTan[a*x])/(2*a^4*c^2*(1 + a^2*x^2)) - (3*ArcTa
n[a*x]^2)/(4*a^5*c^2) - Log[1 + a^2*x^2]/(2*a^5*c^2)

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Rubi [A]  time = 0.180487, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4964, 4916, 4846, 260, 4884, 4934} \[ \frac{1}{4 a^5 c^2 \left (a^2 x^2+1\right )}-\frac{\log \left (a^2 x^2+1\right )}{2 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{4 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{a^4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]

[Out]

1/(4*a^5*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(a^4*c^2) + (x*ArcTan[a*x])/(2*a^4*c^2*(1 + a^2*x^2)) - (3*ArcTa
n[a*x]^2)/(4*a^5*c^2) - Log[1 + a^2*x^2]/(2*a^5*c^2)

Rule 4964

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

Rule 4916

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

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]

Rule 4884

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

Rule 4934

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^2*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q
+ 1))/(4*c^3*d*(q + 1)^2), x] + (-Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x],
x] + Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && E
qQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]

Rubi steps

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

Mathematica [A]  time = 0.0598482, size = 79, normalized size = 0.82 \[ \frac{-2 \left (a^2 x^2+1\right ) \log \left (a^2 x^2+1\right )-3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+\left (4 a^3 x^3+6 a x\right ) \tan ^{-1}(a x)+1}{4 a^5 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]

[Out]

(1 + (6*a*x + 4*a^3*x^3)*ArcTan[a*x] - 3*(1 + a^2*x^2)*ArcTan[a*x]^2 - 2*(1 + a^2*x^2)*Log[1 + a^2*x^2])/(4*a^
5*c^2*(1 + a^2*x^2))

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Maple [A]  time = 0.044, size = 89, normalized size = 0.9 \begin{align*}{\frac{1}{4\,{a}^{5}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{x\arctan \left ( ax \right ) }{{a}^{4}{c}^{2}}}+{\frac{x\arctan \left ( ax \right ) }{2\,{a}^{4}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{a}^{5}{c}^{2}}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{a}^{5}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arctan(a*x)/(a^2*c*x^2+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.57983, size = 154, normalized size = 1.6 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{a^{6} c^{2} x^{2} + a^{4} c^{2}} + \frac{2 \, x}{a^{4} c^{2}} - \frac{3 \, \arctan \left (a x\right )}{a^{5} c^{2}}\right )} \arctan \left (a x\right ) + \frac{{\left (3 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 2 \,{\left (a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) + 1\right )} a}{4 \,{\left (a^{8} c^{2} x^{2} + a^{6} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(a*x)/(a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

1/2*(x/(a^6*c^2*x^2 + a^4*c^2) + 2*x/(a^4*c^2) - 3*arctan(a*x)/(a^5*c^2))*arctan(a*x) + 1/4*(3*(a^2*x^2 + 1)*a
rctan(a*x)^2 - 2*(a^2*x^2 + 1)*log(a^2*x^2 + 1) + 1)*a/(a^8*c^2*x^2 + a^6*c^2)

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Fricas [A]  time = 1.63435, size = 185, normalized size = 1.93 \begin{align*} -\frac{3 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 2 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) + 2 \,{\left (a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) - 1}{4 \,{\left (a^{7} c^{2} x^{2} + a^{5} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(a*x)/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/4*(3*(a^2*x^2 + 1)*arctan(a*x)^2 - 2*(2*a^3*x^3 + 3*a*x)*arctan(a*x) + 2*(a^2*x^2 + 1)*log(a^2*x^2 + 1) - 1
)/(a^7*c^2*x^2 + a^5*c^2)

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Sympy [A]  time = 3.1785, size = 291, normalized size = 3.03 \begin{align*} \begin{cases} \frac{12 a^{3} x^{3} \operatorname{atan}{\left (a x \right )}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} - \frac{6 a^{2} x^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} - \frac{9 a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} - \frac{a^{2} x^{2}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} + \frac{18 a x \operatorname{atan}{\left (a x \right )}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} - \frac{6 \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} - \frac{9 \operatorname{atan}^{2}{\left (a x \right )}}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} + \frac{2}{12 a^{7} c^{2} x^{2} + 12 a^{5} c^{2}} & \text{for}\: c \neq 0 \\\tilde{\infty } \left (\frac{x^{5} \operatorname{atan}{\left (a x \right )}}{5} - \frac{x^{4}}{20 a} + \frac{x^{2}}{10 a^{3}} - \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*atan(a*x)/(a**2*c*x**2+c)**2,x)

[Out]

Piecewise((12*a**3*x**3*atan(a*x)/(12*a**7*c**2*x**2 + 12*a**5*c**2) - 6*a**2*x**2*log(x**2 + a**(-2))/(12*a**
7*c**2*x**2 + 12*a**5*c**2) - 9*a**2*x**2*atan(a*x)**2/(12*a**7*c**2*x**2 + 12*a**5*c**2) - a**2*x**2/(12*a**7
*c**2*x**2 + 12*a**5*c**2) + 18*a*x*atan(a*x)/(12*a**7*c**2*x**2 + 12*a**5*c**2) - 6*log(x**2 + a**(-2))/(12*a
**7*c**2*x**2 + 12*a**5*c**2) - 9*atan(a*x)**2/(12*a**7*c**2*x**2 + 12*a**5*c**2) + 2/(12*a**7*c**2*x**2 + 12*
a**5*c**2), Ne(c, 0)), (zoo*(x**5*atan(a*x)/5 - x**4/(20*a) + x**2/(10*a**3) - log(a**2*x**2 + 1)/(10*a**5)),
True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(a*x)/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

integrate(x^4*arctan(a*x)/(a^2*c*x^2 + c)^2, x)