3.104 \(\int \frac{\cot ^{-1}(a+b x)}{x^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b \log (x)}{a^2+1}+\frac{b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}+\frac{a b \tan ^{-1}(a+b x)}{a^2+1}-\frac{\cot ^{-1}(a+b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0393536, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5046, 371, 706, 31, 635, 203, 260} \[ -\frac{b \log (x)}{a^2+1}+\frac{b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}+\frac{a b \tan ^{-1}(a+b x)}{a^2+1}-\frac{\cot ^{-1}(a+b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a + b*x]/x^2,x]

[Out]

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

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]

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 706

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

Rule 31

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

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 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]

Rubi steps

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

Mathematica [C]  time = 0.0521144, size = 66, normalized size = 1.06 \[ -\frac{\cot ^{-1}(a+b x)}{x}+\frac{b ((1-i a) \log (-a-b x+i)+(1+i a) \log (a+b x+i)-2 \log (x))}{2 \left (a^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a + b*x]/x^2,x]

[Out]

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

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Maple [A]  time = 0.049, size = 63, normalized size = 1. \begin{align*} -{\frac{{\rm arccot} \left (bx+a\right )}{x}}+{\frac{b\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ) }{2\,{a}^{2}+2}}+{\frac{ab\arctan \left ( bx+a \right ) }{{a}^{2}+1}}-{\frac{b\ln \left ( bx \right ) }{{a}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(b*x+a)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.47844, size = 104, normalized size = 1.68 \begin{align*} \frac{1}{2} \, b{\left (\frac{2 \, a \arctan \left (\frac{b^{2} x + a b}{b}\right )}{a^{2} + 1} + \frac{\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{2} + 1} - \frac{2 \, \log \left (x\right )}{a^{2} + 1}\right )} - \frac{\operatorname{arccot}\left (b x + a\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(b*x+a)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.31603, size = 174, normalized size = 2.81 \begin{align*} \frac{2 \, a b x \arctan \left (b x + a\right ) + b x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, b x \log \left (x\right ) - 2 \,{\left (a^{2} + 1\right )} \operatorname{arccot}\left (b x + a\right )}{2 \,{\left (a^{2} + 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(b*x+a)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 9.50362, size = 330, normalized size = 5.32 \begin{align*} \begin{cases} - \frac{2 i b^{2} x^{2} \operatorname{acot}{\left (b x - i \right )}}{4 b x^{2} - 8 i x} + \frac{b^{2} x^{2}}{4 b x^{2} - 8 i x} - \frac{8 b x \operatorname{acot}{\left (b x - i \right )}}{4 b x^{2} - 8 i x} + \frac{8 i \operatorname{acot}{\left (b x - i \right )}}{4 b x^{2} - 8 i x} + \frac{4}{4 b x^{2} - 8 i x} & \text{for}\: a = - i \\\frac{2 i b^{2} x^{2} \operatorname{acot}{\left (b x + i \right )}}{4 b x^{2} + 8 i x} + \frac{b^{2} x^{2}}{4 b x^{2} + 8 i x} - \frac{8 b x \operatorname{acot}{\left (b x + i \right )}}{4 b x^{2} + 8 i x} - \frac{8 i \operatorname{acot}{\left (b x + i \right )}}{4 b x^{2} + 8 i x} + \frac{4}{4 b x^{2} + 8 i x} & \text{for}\: a = i \\- \frac{2 a^{2} \operatorname{acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac{2 a b x \operatorname{acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac{2 b x \log{\left (x \right )}}{2 a^{2} x + 2 x} + \frac{b x \log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 a^{2} x + 2 x} - \frac{2 \operatorname{acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(b*x+a)/x**2,x)

[Out]

Piecewise((-2*I*b**2*x**2*acot(b*x - I)/(4*b*x**2 - 8*I*x) + b**2*x**2/(4*b*x**2 - 8*I*x) - 8*b*x*acot(b*x - I
)/(4*b*x**2 - 8*I*x) + 8*I*acot(b*x - I)/(4*b*x**2 - 8*I*x) + 4/(4*b*x**2 - 8*I*x), Eq(a, -I)), (2*I*b**2*x**2
*acot(b*x + I)/(4*b*x**2 + 8*I*x) + b**2*x**2/(4*b*x**2 + 8*I*x) - 8*b*x*acot(b*x + I)/(4*b*x**2 + 8*I*x) - 8*
I*acot(b*x + I)/(4*b*x**2 + 8*I*x) + 4/(4*b*x**2 + 8*I*x), Eq(a, I)), (-2*a**2*acot(a + b*x)/(2*a**2*x + 2*x)
- 2*a*b*x*acot(a + b*x)/(2*a**2*x + 2*x) - 2*b*x*log(x)/(2*a**2*x + 2*x) + b*x*log(a**2 + 2*a*b*x + b**2*x**2
+ 1)/(2*a**2*x + 2*x) - 2*acot(a + b*x)/(2*a**2*x + 2*x), True))

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Giac [A]  time = 1.11117, size = 97, normalized size = 1.56 \begin{align*} \frac{1}{2} \, b{\left (\frac{2 \, a \arctan \left (b x + a\right )}{a^{2} + 1} + \frac{\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{2} + 1} - \frac{2 \, \log \left ({\left | x \right |}\right )}{a^{2} + 1}\right )} - \frac{\arctan \left (\frac{1}{b x + a}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(b*x+a)/x^2,x, algorithm="giac")

[Out]

1/2*b*(2*a*arctan(b*x + a)/(a^2 + 1) + log(b^2*x^2 + 2*a*b*x + a^2 + 1)/(a^2 + 1) - 2*log(abs(x))/(a^2 + 1)) -
arctan(1/(b*x + a))/x