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

Optimal. Leaf size=85 \[ -\frac{4}{3} a^3 c^2 \log \left (a^2 x^2+1\right )+\frac{5}{3} a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{a c^2}{6 x^2}-\frac{c^2 \tan ^{-1}(a x)}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.124028, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4948, 4846, 260, 4852, 266, 44, 36, 29, 31} \[ -\frac{4}{3} a^3 c^2 \log \left (a^2 x^2+1\right )+\frac{5}{3} a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{a c^2}{6 x^2}-\frac{c^2 \tan ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4948

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

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 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0548929, size = 68, normalized size = 0.8 \[ \frac{c^2 \left (a x \left (10 a^2 x^2 \log (x)-8 a^2 x^2 \log \left (a^2 x^2+1\right )-1\right )+2 \left (3 a^4 x^4-6 a^2 x^2-1\right ) \tan ^{-1}(a x)\right )}{6 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(2*(-1 - 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + a*x*(-1 + 10*a^2*x^2*Log[x] - 8*a^2*x^2*Log[1 + a^2*x^2])))
/(6*x^3)

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Maple [A]  time = 0.032, size = 80, normalized size = 0.9 \begin{align*}{a}^{4}{c}^{2}x\arctan \left ( ax \right ) -2\,{\frac{{a}^{2}{c}^{2}\arctan \left ( ax \right ) }{x}}-{\frac{{c}^{2}\arctan \left ( ax \right ) }{3\,{x}^{3}}}-{\frac{4\,{a}^{3}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{{c}^{2}a}{6\,{x}^{2}}}+{\frac{5\,{a}^{3}{c}^{2}\ln \left ( ax \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00723, size = 103, normalized size = 1.21 \begin{align*} -\frac{1}{6} \,{\left (8 \, a^{2} c^{2} \log \left (a^{2} x^{2} + 1\right ) - 10 \, a^{2} c^{2} \log \left (x\right ) + \frac{c^{2}}{x^{2}}\right )} a + \frac{1}{3} \,{\left (3 \, a^{4} c^{2} x - \frac{6 \, a^{2} c^{2} x^{2} + c^{2}}{x^{3}}\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)^2*arctan(a*x)/x^4,x, algorithm="maxima")

[Out]

-1/6*(8*a^2*c^2*log(a^2*x^2 + 1) - 10*a^2*c^2*log(x) + c^2/x^2)*a + 1/3*(3*a^4*c^2*x - (6*a^2*c^2*x^2 + c^2)/x
^3)*arctan(a*x)

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Fricas [A]  time = 1.72893, size = 177, normalized size = 2.08 \begin{align*} -\frac{8 \, a^{3} c^{2} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 10 \, a^{3} c^{2} x^{3} \log \left (x\right ) + a c^{2} x - 2 \,{\left (3 \, a^{4} c^{2} x^{4} - 6 \, a^{2} c^{2} x^{2} - c^{2}\right )} \arctan \left (a x\right )}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(8*a^3*c^2*x^3*log(a^2*x^2 + 1) - 10*a^3*c^2*x^3*log(x) + a*c^2*x - 2*(3*a^4*c^2*x^4 - 6*a^2*c^2*x^2 - c^
2)*arctan(a*x))/x^3

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Sympy [A]  time = 1.92768, size = 87, normalized size = 1.02 \begin{align*} \begin{cases} a^{4} c^{2} x \operatorname{atan}{\left (a x \right )} + \frac{5 a^{3} c^{2} \log{\left (x \right )}}{3} - \frac{4 a^{3} c^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{3} - \frac{2 a^{2} c^{2} \operatorname{atan}{\left (a x \right )}}{x} - \frac{a c^{2}}{6 x^{2}} - \frac{c^{2} \operatorname{atan}{\left (a x \right )}}{3 x^{3}} & \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)**2*atan(a*x)/x**4,x)

[Out]

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

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Giac [A]  time = 1.12624, size = 120, normalized size = 1.41 \begin{align*} -\frac{4}{3} \, a^{3} c^{2} \log \left (a^{2} x^{2} + 1\right ) + \frac{5}{6} \, a^{3} c^{2} \log \left (x^{2}\right ) + \frac{1}{3} \,{\left (3 \, a^{4} c^{2} x - \frac{6 \, a^{2} c^{2} x^{2} + c^{2}}{x^{3}}\right )} \arctan \left (a x\right ) - \frac{5 \, a^{3} c^{2} x^{2} + a c^{2}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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