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

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

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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])

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 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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.50, size = 100, normalized size = 0.86 \[ -\frac {a^{5} c^{3} x^{5} + 16 \, a^{3} c^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{3} c^{3} x^{3} \log \relax (x) + a c^{3} x - 2 \, {\left (a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} - 9 \, a^{2} c^{3} x^{2} - c^{3}\right )} \arctan \left (a x\right )}{6 \, x^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.04, size = 107, normalized size = 0.92 \[ \frac {a^{6} c^{3} x^{3} \arctan \left (a x \right )}{3}+3 a^{4} c^{3} x \arctan \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{3 x^{3}}-\frac {3 a^{2} c^{3} \arctan \left (a x \right )}{x}-\frac {a^{5} c^{3} x^{2}}{6}-\frac {a \,c^{3}}{6 x^{2}}+\frac {8 a^{3} c^{3} \ln \left (a x \right )}{3}-\frac {8 a^{3} c^{3} \ln \left (a^{2} x^{2}+1\right )}{3} \]

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

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maxima [A]  time = 0.33, size = 96, normalized size = 0.83 \[ -\frac {1}{6} \, {\left (a^{4} c^{3} x^{2} + 16 \, a^{2} c^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{2} c^{3} \log \relax (x) + \frac {c^{3}}{x^{2}}\right )} a + \frac {1}{3} \, {\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac {9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) \]

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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mupad [B]  time = 0.54, size = 97, normalized size = 0.84 \[ -\frac {c^3\,\left (2\,\mathrm {atan}\left (a\,x\right )+a\,x-a^3\,x^3+a^5\,x^5+18\,a^2\,x^2\,\mathrm {atan}\left (a\,x\right )-18\,a^4\,x^4\,\mathrm {atan}\left (a\,x\right )-2\,a^6\,x^6\,\mathrm {atan}\left (a\,x\right )+16\,a^3\,x^3\,\ln \left (a^2\,x^2+1\right )-16\,a^3\,x^3\,\ln \relax (x)\right )}{6\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atan(a*x)*(c + a^2*c*x^2)^3)/x^4,x)

[Out]

-(c^3*(2*atan(a*x) + a*x - a^3*x^3 + a^5*x^5 + 18*a^2*x^2*atan(a*x) - 18*a^4*x^4*atan(a*x) - 2*a^6*x^6*atan(a*
x) + 16*a^3*x^3*log(a^2*x^2 + 1) - 16*a^3*x^3*log(x)))/(6*x^3)

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sympy [A]  time = 2.17, size = 117, normalized size = 1.01 \[ \begin {cases} \frac {a^{6} c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {a^{5} c^{3} x^{2}}{6} + 3 a^{4} c^{3} x \operatorname {atan}{\left (a x \right )} + \frac {8 a^{3} c^{3} \log {\relax (x )}}{3} - \frac {8 a^{3} c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3} - \frac {3 a^{2} c^{3} \operatorname {atan}{\left (a x \right )}}{x} - \frac {a c^{3}}{6 x^{2}} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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