3.172 \(\int \frac{(c+a^2 c x^2)^3 \tan ^{-1}(a x)}{x^4} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{6} a^5 c^3 x^2-\frac{8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{a c^3}{6 x^2}-\frac{c^3 \tan ^{-1}(a x)}{3 x^3} \]

[Out]

-(a*c^3)/(6*x^2) - (a^5*c^3*x^2)/6 - (c^3*ArcTan[a*x])/(3*x^3) - (3*a^2*c^3*ArcTan[a*x])/x + 3*a^4*c^3*x*ArcTa
n[a*x] + (a^6*c^3*x^3*ArcTan[a*x])/3 + (8*a^3*c^3*Log[x])/3 - (8*a^3*c^3*Log[1 + a^2*x^2])/3

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Rubi [A]  time = 0.157906, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4948, 4846, 260, 4852, 266, 44, 36, 29, 31, 43} \[ -\frac{1}{6} a^5 c^3 x^2-\frac{8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{a c^3}{6 x^2}-\frac{c^3 \tan ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4,x]

[Out]

-(a*c^3)/(6*x^2) - (a^5*c^3*x^2)/6 - (c^3*ArcTan[a*x])/(3*x^3) - (3*a^2*c^3*ArcTan[a*x])/x + 3*a^4*c^3*x*ArcTa
n[a*x] + (a^6*c^3*x^3*ArcTan[a*x])/3 + (8*a^3*c^3*Log[x])/3 - (8*a^3*c^3*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]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)}{x^4} \, dx &=\int \left (3 a^4 c^3 \tan ^{-1}(a x)+\frac{c^3 \tan ^{-1}(a x)}{x^4}+\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x^2}+a^6 c^3 x^2 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)}{x^4} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^2} \, dx+\left (3 a^4 c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{1}{3} \left (a c^3\right ) \int \frac{1}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (3 a^3 c^3\right ) \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^5 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^7 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac{3}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{2} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{6} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac{3}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+\frac{1}{6} \left (a c^3\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 )+\frac{1}{2} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (3 a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{6} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{a c^3}{6 x^2}-\frac{1}{6} a^5 c^3 x^2-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)-\frac{8}{3} a^3 c^3 \log \left (1+a^2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0706868, size = 83, normalized size = 0.72 \[ \frac{c^3 \left (2 \left (a^6 x^6+9 a^4 x^4-9 a^2 x^2-1\right ) \tan ^{-1}(a x)-a x \left (a^4 x^4-16 a^2 x^2 \log (x)+16 a^2 x^2 \log \left (a^2 x^2+1\right )+1\right )\right )}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4,x]

[Out]

(c^3*(2*(-1 - 9*a^2*x^2 + 9*a^4*x^4 + a^6*x^6)*ArcTan[a*x] - a*x*(1 + a^4*x^4 - 16*a^2*x^2*Log[x] + 16*a^2*x^2
*Log[1 + a^2*x^2])))/(6*x^3)

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Maple [A]  time = 0.033, size = 107, normalized size = 0.9 \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{3}\arctan \left ( ax \right ) }{3}}+3\,{a}^{4}{c}^{3}x\arctan \left ( ax \right ) -3\,{\frac{{a}^{2}{c}^{3}\arctan \left ( ax \right ) }{x}}-{\frac{{c}^{3}\arctan \left ( ax \right ) }{3\,{x}^{3}}}-{\frac{{a}^{5}{c}^{3}{x}^{2}}{6}}-{\frac{8\,{a}^{3}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{a{c}^{3}}{6\,{x}^{2}}}+{\frac{8\,{a}^{3}{c}^{3}\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)^3*arctan(a*x)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.12865, size = 130, normalized size = 1.12 \begin{align*} -\frac{1}{6} \,{\left (a^{4} c^{3} x^{2} + 16 \, a^{2} c^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{2} c^{3} \log \left (x\right ) + \frac{c^{3}}{x^{2}}\right )} a + \frac{1}{3} \,{\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac{9 \, a^{2} c^{3} x^{2} + c^{3}}{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)^3*arctan(a*x)/x^4,x, algorithm="maxima")

[Out]

-1/6*(a^4*c^3*x^2 + 16*a^2*c^3*log(a^2*x^2 + 1) - 16*a^2*c^3*log(x) + c^3/x^2)*a + 1/3*(a^6*c^3*x^3 + 9*a^4*c^
3*x - (9*a^2*c^3*x^2 + c^3)/x^3)*arctan(a*x)

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Fricas [A]  time = 1.74709, size = 216, normalized size = 1.86 \begin{align*} -\frac{a^{5} c^{3} x^{5} + 16 \, a^{3} c^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{3} c^{3} x^{3} \log \left (x\right ) + a c^{3} x - 2 \,{\left (a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} - 9 \, a^{2} c^{3} x^{2} - c^{3}\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)^3*arctan(a*x)/x^4,x, algorithm="fricas")

[Out]

-1/6*(a^5*c^3*x^5 + 16*a^3*c^3*x^3*log(a^2*x^2 + 1) - 16*a^3*c^3*x^3*log(x) + a*c^3*x - 2*(a^6*c^3*x^6 + 9*a^4
*c^3*x^4 - 9*a^2*c^3*x^2 - c^3)*arctan(a*x))/x^3

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Sympy [A]  time = 2.94068, size = 117, normalized size = 1.01 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{a^{5} c^{3} x^{2}}{6} + 3 a^{4} c^{3} x \operatorname{atan}{\left (a x \right )} + \frac{8 a^{3} c^{3} \log{\left (x \right )}}{3} - \frac{8 a^{3} c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{3} - \frac{3 a^{2} c^{3} \operatorname{atan}{\left (a x \right )}}{x} - \frac{a c^{3}}{6 x^{2}} - \frac{c^{3} \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)**3*atan(a*x)/x**4,x)

[Out]

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

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Giac [A]  time = 1.18822, size = 149, normalized size = 1.28 \begin{align*} -\frac{1}{6} \, a^{5} c^{3} x^{2} - \frac{8}{3} \, a^{3} c^{3} \log \left (a^{2} x^{2} + 1\right ) + \frac{4}{3} \, a^{3} c^{3} \log \left (x^{2}\right ) + \frac{1}{3} \,{\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac{9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) - \frac{8 \, a^{3} c^{3} x^{2} + a c^{3}}{6 \, x^{2}} \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^4,x, algorithm="giac")

[Out]

-1/6*a^5*c^3*x^2 - 8/3*a^3*c^3*log(a^2*x^2 + 1) + 4/3*a^3*c^3*log(x^2) + 1/3*(a^6*c^3*x^3 + 9*a^4*c^3*x - (9*a
^2*c^3*x^2 + c^3)/x^3)*arctan(a*x) - 1/6*(8*a^3*c^3*x^2 + a*c^3)/x^2