3.106 \(\int \frac {\cot ^{-1}(a+b x)}{x^4} \, dx\)

Optimal. Leaf size=129 \[ \frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (a^2+1\right )^3}-\frac {\left (1-3 a^2\right ) b^3 \log \left ((a+b x)^2+1\right )}{6 \left (a^2+1\right )^3}-\frac {a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (a^2+1\right )^3}-\frac {2 a b^2}{3 \left (a^2+1\right )^2 x}+\frac {b}{6 \left (a^2+1\right ) x^2}-\frac {\cot ^{-1}(a+b x)}{3 x^3} \]

[Out]

1/6*b/(a^2+1)/x^2-2/3*a*b^2/(a^2+1)^2/x-1/3*arccot(b*x+a)/x^3-1/3*a*(-a^2+3)*b^3*arctan(b*x+a)/(a^2+1)^3+1/3*(
-3*a^2+1)*b^3*ln(x)/(a^2+1)^3-1/6*(-3*a^2+1)*b^3*ln(1+(b*x+a)^2)/(a^2+1)^3

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Rubi [A]  time = 0.11, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5046, 371, 710, 801, 635, 203, 260} \[ -\frac {2 a b^2}{3 \left (a^2+1\right )^2 x}+\frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (a^2+1\right )^3}-\frac {\left (1-3 a^2\right ) b^3 \log \left ((a+b x)^2+1\right )}{6 \left (a^2+1\right )^3}-\frac {a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (a^2+1\right )^3}+\frac {b}{6 \left (a^2+1\right ) x^2}-\frac {\cot ^{-1}(a+b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a + b*x]/x^4,x]

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 5046

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

Rubi steps

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

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Mathematica [C]  time = 0.16, size = 126, normalized size = 0.98 \[ \frac {2 \left (1-3 a^2\right ) b^3 x^3 \log (x)+(a-i) b x \left ((a+i) \left (a^2-4 a b x+1\right )+i (a-i)^2 b^2 x^2 \log (a+b x+i)\right )-2 \left (a^2+1\right )^3 \cot ^{-1}(a+b x)+(-1+i a)^3 b^3 x^3 \log (-a-b x+i)}{6 \left (a^2+1\right )^3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a + b*x]/x^4,x]

[Out]

(-2*(1 + a^2)^3*ArcCot[a + b*x] + 2*(1 - 3*a^2)*b^3*x^3*Log[x] + (-1 + I*a)^3*b^3*x^3*Log[I - a - b*x] + (-I +
a)*b*x*((I + a)*(1 + a^2 - 4*a*b*x) + I*(-I + a)^2*b^2*x^2*Log[I + a + b*x]))/(6*(1 + a^2)^3*x^3)

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fricas [A]  time = 0.59, size = 142, normalized size = 1.10 \[ \frac {2 \, {\left (a^{3} - 3 \, a\right )} b^{3} x^{3} \arctan \left (b x + a\right ) + {\left (3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, {\left (3 \, a^{2} - 1\right )} b^{3} x^{3} \log \relax (x) - 4 \, {\left (a^{3} + a\right )} b^{2} x^{2} + {\left (a^{4} + 2 \, a^{2} + 1\right )} b x - 2 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} \operatorname {arccot}\left (b x + a\right )}{6 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(b*x+a)/x^4,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 2.61, size = 3449, normalized size = 26.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(b*x+a)/x^4,x, algorithm="giac")

[Out]

-1/6*(24*a^5*b^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^4 + 12*a^4*b^2*arctan(1/(b*x + a))*tan(1/2*a
rctan(1/(b*x + a)))^5 + 2*a^3*b^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^6 + 24*a^5*b^2*log(4*(4*a^2
*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*ta
n(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2
*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^3 + 36*a^4*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x +
a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a)
)) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2
+ 1))*tan(1/2*arctan(1/(b*x + a)))^4 + 18*a^3*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*ar
ctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/
(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(
b*x + a)))^5 + 3*a^2*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + ta
n(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1
/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^6 - 24*a^5*b^2
*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^2 - 120*a^4*b^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x +
a)))^3 + 24*a^4*b^2*tan(1/2*arctan(1/(b*x + a)))^4 - 78*a^3*b^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)
))^4 + 22*a^3*b^2*tan(1/2*arctan(1/(b*x + a)))^5 - 36*a^2*b^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))
^5 + 5*a^2*b^2*tan(1/2*arctan(1/(b*x + a)))^6 - 6*a*b^2*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^6 - 3
6*a^4*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/
(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*
x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^2 - 44*a^3*b^2*log(4*(4*a^2*t
an(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(
1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*a
rctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^3 - 21*a^2*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a
)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a)))
- 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 +
1))*tan(1/2*arctan(1/(b*x + a)))^4 - 6*a*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(
1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x
+ a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x +
a)))^5 - b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arcta
n(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1
/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^6 + 12*a^4*b^2*arctan(1/(
b*x + a))*tan(1/2*arctan(1/(b*x + a))) - 24*a^4*b^2*tan(1/2*arctan(1/(b*x + a)))^2 + 78*a^3*b^2*arctan(1/(b*x
+ a))*tan(1/2*arctan(1/(b*x + a)))^2 - 100*a^3*b^2*tan(1/2*arctan(1/(b*x + a)))^3 + 24*a^2*b^2*arctan(1/(b*x +
a))*tan(1/2*arctan(1/(b*x + a)))^3 - 67*a^2*b^2*tan(1/2*arctan(1/(b*x + a)))^4 + 18*a*b^2*arctan(1/(b*x + a))
*tan(1/2*arctan(1/(b*x + a)))^4 - 14*a*b^2*tan(1/2*arctan(1/(b*x + a)))^5 - b^2*tan(1/2*arctan(1/(b*x + a)))^6
+ 18*a^3*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arcta
n(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1
/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a))) + 21*a^2*b^2*log(4*(4*a^2
*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*ta
n(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2
*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^2 + 12*a*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a
)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a)))
- 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 +
1))*tan(1/2*arctan(1/(b*x + a)))^3 + 3*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/
(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x +
a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a
)))^4 - 2*a^3*b^2*arctan(1/(b*x + a)) + 22*a^3*b^2*tan(1/2*arctan(1/(b*x + a))) - 36*a^2*b^2*arctan(1/(b*x + a
))*tan(1/2*arctan(1/(b*x + a))) + 67*a^2*b^2*tan(1/2*arctan(1/(b*x + a)))^2 - 18*a*b^2*arctan(1/(b*x + a))*tan
(1/2*arctan(1/(b*x + a)))^2 + 20*a*b^2*tan(1/2*arctan(1/(b*x + a)))^3 - 16*b^2*arctan(1/(b*x + a))*tan(1/2*arc
tan(1/(b*x + a)))^3 - b^2*tan(1/2*arctan(1/(b*x + a)))^4 - 3*a^2*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))
^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) -
2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))
- 6*a*b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1
/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b
*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a))) - 3*b^2*log(4*(4*a^2*tan(1/2
*arctan(1/(b*x + a)))^2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*ar
ctan(1/(b*x + a))) - 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(
1/(b*x + a)))^2 + 1))*tan(1/2*arctan(1/(b*x + a)))^2 - 5*a^2*b^2 + 6*a*b^2*arctan(1/(b*x + a)) - 14*a*b^2*tan(
1/2*arctan(1/(b*x + a))) + b^2*tan(1/2*arctan(1/(b*x + a)))^2 + b^2*log(4*(4*a^2*tan(1/2*arctan(1/(b*x + a)))^
2 + 4*a*tan(1/2*arctan(1/(b*x + a)))^3 + tan(1/2*arctan(1/(b*x + a)))^4 - 4*a*tan(1/2*arctan(1/(b*x + a))) - 2
*tan(1/2*arctan(1/(b*x + a)))^2 + 1)/(tan(1/2*arctan(1/(b*x + a)))^4 + 2*tan(1/2*arctan(1/(b*x + a)))^2 + 1))
+ b^2)*b/(8*a^9*tan(1/2*arctan(1/(b*x + a)))^3 + 12*a^8*tan(1/2*arctan(1/(b*x + a)))^4 + 6*a^7*tan(1/2*arctan(
1/(b*x + a)))^5 + a^6*tan(1/2*arctan(1/(b*x + a)))^6 - 12*a^8*tan(1/2*arctan(1/(b*x + a)))^2 + 12*a^7*tan(1/2*
arctan(1/(b*x + a)))^3 + 33*a^6*tan(1/2*arctan(1/(b*x + a)))^4 + 18*a^5*tan(1/2*arctan(1/(b*x + a)))^5 + 3*a^4
*tan(1/2*arctan(1/(b*x + a)))^6 + 6*a^7*tan(1/2*arctan(1/(b*x + a))) - 33*a^6*tan(1/2*arctan(1/(b*x + a)))^2 -
12*a^5*tan(1/2*arctan(1/(b*x + a)))^3 + 27*a^4*tan(1/2*arctan(1/(b*x + a)))^4 + 18*a^3*tan(1/2*arctan(1/(b*x
+ a)))^5 + 3*a^2*tan(1/2*arctan(1/(b*x + a)))^6 - a^6 + 18*a^5*tan(1/2*arctan(1/(b*x + a))) - 27*a^4*tan(1/2*a
rctan(1/(b*x + a)))^2 - 28*a^3*tan(1/2*arctan(1/(b*x + a)))^3 + 3*a^2*tan(1/2*arctan(1/(b*x + a)))^4 + 6*a*tan
(1/2*arctan(1/(b*x + a)))^5 + tan(1/2*arctan(1/(b*x + a)))^6 - 3*a^4 + 18*a^3*tan(1/2*arctan(1/(b*x + a))) - 3
*a^2*tan(1/2*arctan(1/(b*x + a)))^2 - 12*a*tan(1/2*arctan(1/(b*x + a)))^3 - 3*tan(1/2*arctan(1/(b*x + a)))^4 -
3*a^2 + 6*a*tan(1/2*arctan(1/(b*x + a))) + 3*tan(1/2*arctan(1/(b*x + a)))^2 - 1)

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maple [A]  time = 0.05, size = 164, normalized size = 1.27 \[ -\frac {\mathrm {arccot}\left (b x +a \right )}{3 x^{3}}+\frac {b}{6 \left (a^{2}+1\right ) x^{2}}-\frac {b^{3} \ln \left (b x \right ) a^{2}}{\left (a^{2}+1\right )^{3}}+\frac {b^{3} \ln \left (b x \right )}{3 \left (a^{2}+1\right )^{3}}-\frac {2 a \,b^{2}}{3 \left (a^{2}+1\right )^{2} x}+\frac {b^{3} \ln \left (1+\left (b x +a \right )^{2}\right ) a^{2}}{2 \left (a^{2}+1\right )^{3}}-\frac {b^{3} \ln \left (1+\left (b x +a \right )^{2}\right )}{6 \left (a^{2}+1\right )^{3}}+\frac {b^{3} \arctan \left (b x +a \right ) a^{3}}{3 \left (a^{2}+1\right )^{3}}-\frac {b^{3} \arctan \left (b x +a \right ) a}{\left (a^{2}+1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(b*x+a)/x^4,x)

[Out]

-1/3*arccot(b*x+a)/x^3+1/6*b/(a^2+1)/x^2-b^3/(a^2+1)^3*ln(b*x)*a^2+1/3*b^3/(a^2+1)^3*ln(b*x)-2/3*a*b^2/(a^2+1)
^2/x+1/2*b^3/(a^2+1)^3*ln(1+(b*x+a)^2)*a^2-1/6*b^3/(a^2+1)^3*ln(1+(b*x+a)^2)+1/3*b^3/(a^2+1)^3*arctan(b*x+a)*a
^3-b^3/(a^2+1)^3*arctan(b*x+a)*a

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maxima [A]  time = 0.42, size = 165, normalized size = 1.28 \[ \frac {1}{6} \, {\left (\frac {2 \, {\left (a^{3} - 3 \, a\right )} b^{2} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} + \frac {{\left (3 \, a^{2} - 1\right )} b^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {2 \, {\left (3 \, a^{2} - 1\right )} b^{2} \log \relax (x)}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {4 \, a b x - a^{2} - 1}{{\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}}\right )} b - \frac {\operatorname {arccot}\left (b x + a\right )}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(b*x+a)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.24, size = 285, normalized size = 2.21 \[ \frac {\ln \relax (x)\,\left (\frac {b^3}{3}-a^2\,b^3\right )}{a^6+3\,a^4+3\,a^2+1}-\frac {\mathrm {acot}\left (a+b\,x\right )\,\left (\frac {a^2}{3}+\frac {1}{3}\right )-\frac {b\,x}{6}+\frac {b^2\,x^2\,\mathrm {acot}\left (a+b\,x\right )}{3}-\frac {x^3\,\left (b^3-7\,a^2\,b^3\right )}{6\,\left (a^4+2\,a^2+1\right )}+\frac {a\,b^2\,x^2}{3\,\left (a^2+1\right )}+\frac {2\,a\,b^4\,x^4}{3\,{\left (a^2+1\right )}^2}+\frac {2\,a\,b\,x\,\mathrm {acot}\left (a+b\,x\right )}{3}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5+x^3}+\frac {b^3\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )\,\left (3\,a^2-1\right )}{6\,\left (a^6+3\,a^4+3\,a^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {2\,x\,b^2+2\,a\,b}{2\,\sqrt {b^2\,\left (a^2+1\right )-a^2\,b^2}}\right )\,\left (a^2-3\right )\,{\left (b^2\right )}^{3/2}}{3\,\left (a^6+3\,a^4+3\,a^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a + b*x)/x^4,x)

[Out]

(log(x)*(b^3/3 - a^2*b^3))/(3*a^2 + 3*a^4 + a^6 + 1) - (acot(a + b*x)*(a^2/3 + 1/3) - (b*x)/6 + (b^2*x^2*acot(
a + b*x))/3 - (x^3*(b^3 - 7*a^2*b^3))/(6*(2*a^2 + a^4 + 1)) + (a*b^2*x^2)/(3*(a^2 + 1)) + (2*a*b^4*x^4)/(3*(a^
2 + 1)^2) + (2*a*b*x*acot(a + b*x))/3)/(x^3 + a^2*x^3 + b^2*x^5 + 2*a*b*x^4) + (b^3*log(a^2 + b^2*x^2 + 2*a*b*
x + 1)*(3*a^2 - 1))/(6*(3*a^2 + 3*a^4 + a^6 + 1)) + (a*atan((2*a*b + 2*b^2*x)/(2*(b^2*(a^2 + 1) - a^2*b^2)^(1/
2)))*(a^2 - 3)*(b^2)^(3/2))/(3*(3*a^2 + 3*a^4 + a^6 + 1))

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sympy [B]  time = 4.51, size = 760, normalized size = 5.89 \[ \begin {cases} \frac {i b^{3} \operatorname {acot}{\left (b x - i \right )}}{24} - \frac {i b^{2}}{24 x} + \frac {b}{24 x^{2}} - \frac {\operatorname {acot}{\left (b x - i \right )}}{3 x^{3}} + \frac {i}{18 x^{3}} & \text {for}\: a = - i \\- \frac {i b^{3} \operatorname {acot}{\left (b x + i \right )}}{24} + \frac {i b^{2}}{24 x} + \frac {b}{24 x^{2}} - \frac {\operatorname {acot}{\left (b x + i \right )}}{3 x^{3}} - \frac {i}{18 x^{3}} & \text {for}\: a = i \\- \frac {2 a^{6} \operatorname {acot}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {a^{4} b x}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {6 a^{4} \operatorname {acot}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {2 a^{3} b^{3} x^{3} \operatorname {acot}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {4 a^{3} b^{2} x^{2}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {6 a^{2} b^{3} x^{3} \log {\relax (x )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {3 a^{2} b^{3} x^{3} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {2 a^{2} b x}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {6 a^{2} \operatorname {acot}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {6 a b^{3} x^{3} \operatorname {acot}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {4 a b^{2} x^{2}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {2 b^{3} x^{3} \log {\relax (x )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {b^{3} x^{3} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {b x}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {2 \operatorname {acot}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(b*x+a)/x**4,x)

[Out]

Piecewise((I*b**3*acot(b*x - I)/24 - I*b**2/(24*x) + b/(24*x**2) - acot(b*x - I)/(3*x**3) + I/(18*x**3), Eq(a,
-I)), (-I*b**3*acot(b*x + I)/24 + I*b**2/(24*x) + b/(24*x**2) - acot(b*x + I)/(3*x**3) - I/(18*x**3), Eq(a, I
)), (-2*a**6*acot(a + b*x)/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) + a**4*b*x/(6*a**6*x**3 + 18*a
**4*x**3 + 18*a**2*x**3 + 6*x**3) - 6*a**4*acot(a + b*x)/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3)
- 2*a**3*b**3*x**3*acot(a + b*x)/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) - 4*a**3*b**2*x**2/(6*a*
*6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) - 6*a**2*b**3*x**3*log(x)/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**
2*x**3 + 6*x**3) + 3*a**2*b**3*x**3*log(a**2 + 2*a*b*x + b**2*x**2 + 1)/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*
x**3 + 6*x**3) + 2*a**2*b*x/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) - 6*a**2*acot(a + b*x)/(6*a**
6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) + 6*a*b**3*x**3*acot(a + b*x)/(6*a**6*x**3 + 18*a**4*x**3 + 18*
a**2*x**3 + 6*x**3) - 4*a*b**2*x**2/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) + 2*b**3*x**3*log(x)/
(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) - b**3*x**3*log(a**2 + 2*a*b*x + b**2*x**2 + 1)/(6*a**6*x
**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) + b*x/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3) - 2*aco
t(a + b*x)/(6*a**6*x**3 + 18*a**4*x**3 + 18*a**2*x**3 + 6*x**3), True))

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