\(\int \frac {\arctan (a x)}{x^4 (c+a^2 c x^2)} \, dx\) [181]

Optimal result

Integrand size = 20, antiderivative size = 88 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a}{6 c x^2}-\frac {\arctan (a x)}{3 c x^3}+\frac {a^2 \arctan (a x)}{c x}+\frac {a^3 \arctan (a x)^2}{2 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5038, 4946, 272, 46, 36, 29, 31, 5004} \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \arctan (a x)^2}{2 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {a^2 \arctan (a x)}{c x}+\frac {2 a^3 \log \left (a^2 x^2+1\right )}{3 c}-\frac {\arctan (a x)}{3 c x^3}-\frac {a}{6 c x^2} \]

[In]

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

[Out]

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

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 272

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 4946

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

Rule 5004

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 5038

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a}{6 c x^2}-\frac {\arctan (a x)}{3 c x^3}+\frac {a^2 \arctan (a x)}{c x}+\frac {a^3 \arctan (a x)^2}{2 c}-\frac {4 a^3 \log (x)}{3 c}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(-\frac {-3 a^{3} \arctan \left (a x \right )^{2} x^{3}+8 \ln \left (x \right ) a^{3} x^{3}-4 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-2 a^{3} x^{3}-6 a^{2} \arctan \left (a x \right ) x^{2}+a x +2 \arctan \left (a x \right )}{6 c \,x^{3}}\)	\(81\)
derivativedivides	\(a^{3} \left (\frac {\arctan \left (a x \right )^{2}}{c}-\frac {\arctan \left (a x \right )}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )}{c a x}-\frac {-2 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}+4 \ln \left (a x \right )+\frac {3 \arctan \left (a x \right )^{2}}{2}}{3 c}\right )\)	\(85\)
default	\(a^{3} \left (\frac {\arctan \left (a x \right )^{2}}{c}-\frac {\arctan \left (a x \right )}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )}{c a x}-\frac {-2 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}+4 \ln \left (a x \right )+\frac {3 \arctan \left (a x \right )^{2}}{2}}{3 c}\right )\)	\(85\)
parts	\(\frac {a^{3} \arctan \left (a x \right )^{2}}{c}-\frac {\arctan \left (a x \right )}{3 c \,x^{3}}+\frac {a^{2} \arctan \left (a x \right )}{c x}-\frac {a^{3} \left (-2 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}+4 \ln \left (a x \right )\right )+\frac {3 a^{3} \arctan \left (a x \right )^{2}}{2}}{3 c}\)	\(89\)
risch	\(-\frac {a^{3} \ln \left (i a x +1\right )^{2}}{8 c}+\frac {\left (3 a^{3} x^{3} \ln \left (-i a x +1\right )-6 i a^{2} x^{2}+2 i\right ) \ln \left (i a x +1\right )}{12 c \,x^{3}}-\frac {3 a^{3} \ln \left (-i a x +1\right )^{2} x^{3}+32 \ln \left (x \right ) a^{3} x^{3}-16 \ln \left (3 a^{2} x^{2}+3\right ) a^{3} x^{3}-12 i a^{2} x^{2} \ln \left (-i a x +1\right )+4 i \ln \left (-i a x +1\right )+4 a x}{24 c \,x^{3}}\)	\(152\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {3 \, a^{3} x^{3} \arctan \left (a x\right )^{2} + 4 \, a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{3} x^{3} \log \left (x\right ) - a x + 2 \, {\left (3 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{6 \, c x^{3}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\begin {cases} - \frac {4 a^{3} \log {\left (x \right )}}{3 c} + \frac {2 a^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 c} + \frac {a^{3} \operatorname {atan}^{2}{\left (a x \right )}}{2 c} + \frac {a^{2} \operatorname {atan}{\left (a x \right )}}{c x} - \frac {a}{6 c x^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{3 c x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {1}{3} \, {\left (\frac {3 \, a^{3} \arctan \left (a x\right )}{c} + \frac {3 \, a^{2} x^{2} - 1}{c x^{3}}\right )} \arctan \left (a x\right ) - \frac {{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \left (x\right ) + 1\right )} a}{6 \, c x^{2}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {2\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c}-\frac {\mathrm {atan}\left (a\,x\right )}{3\,c\,x^3}-\frac {a}{6\,c\,x^2}-\frac {4\,a^3\,\ln \left (x\right )}{3\,c}+\frac {a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,c}+\frac {a^2\,\mathrm {atan}\left (a\,x\right )}{c\,x} \]

[In]

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

[Out]

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