3.2.4 \(\int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx\) [104]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5154, 378, 720, 31, 649, 209, 266} \begin {gather*} \frac {a b \text {ArcTan}(a+b x)}{a^2+1}-\frac {b \log (x)}{a^2+1}+\frac {b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}-\frac {\cot ^{-1}(a+b x)}{x} \end {gather*}

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 31

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

Rule 209

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

Rule 266

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 378

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 649

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 720

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 5154

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^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 \text {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 \text {Subst}\left (\int \frac {1}{-a+x} \, dx,x,a+b x\right )}{1+a^2}-\frac {b \text {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 \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}+\frac {(a b) \text {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] Result contains complex when optimal does not.
time = 0.04, size = 66, normalized size = 1.06 \begin {gather*} -\frac {\cot ^{-1}(a+b x)}{x}+\frac {b (-2 \log (x)+(1-i a) \log (i-a-b x)+(1+i a) \log (i+a+b x))}{2 \left (1+a^2\right )} \end {gather*}

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))

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 61, normalized size = 0.98

method	result	size
derivativedivides	\(b \left (-\frac {\mathrm {arccot}\left (b x +a \right )}{b x}-\frac {\ln \left (-b x \right )}{a^{2}+1}+\frac {\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}+\arctan \left (b x +a \right ) a}{a^{2}+1}\right )\)	\(61\)
default	\(b \left (-\frac {\mathrm {arccot}\left (b x +a \right )}{b x}-\frac {\ln \left (-b x \right )}{a^{2}+1}+\frac {\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}+\arctan \left (b x +a \right ) a}{a^{2}+1}\right )\)	\(61\)
risch	\(-\frac {i \ln \left (1+i \left (b x +a \right )\right )}{2 x}-\frac {-i a^{2} \ln \left (1-i \left (b x +a \right )\right )-i \ln \left (1-i \left (b x +a \right )\right )+\pi \,a^{2}+\pi +2 \ln \left (x \right ) x b -x b \ln \left (\left (i a b -3 b \right ) x +i a^{2}+3 i-2 a \right )+i x b \ln \left (\left (i a b -3 b \right ) x +i a^{2}+3 i-2 a \right ) a -x b \ln \left (\left (i a b +3 b \right ) x +i a^{2}+3 i+2 a \right )-i x b \ln \left (\left (i a b +3 b \right ) x +i a^{2}+3 i+2 a \right ) a}{2 x \left (a -i\right ) \left (i+a \right )}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 77, normalized size = 1.24 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 3.32, size = 64, normalized size = 1.03 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.80, size = 167, normalized size = 2.69 \begin {gather*} \begin {cases} - \frac {i b \operatorname {acot}{\left (b x - i \right )}}{2} - \frac {\operatorname {acot}{\left (b x - i \right )}}{x} + \frac {i}{2 x} & \text {for}\: a = - i \\\frac {i b \operatorname {acot}{\left (b x + i \right )}}{2} - \frac {\operatorname {acot}{\left (b x + i \right )}}{x} - \frac {i}{2 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*b*acot(b*x - I)/2 - acot(b*x - I)/x + I/(2*x), Eq(a, -I)), (I*b*acot(b*x + I)/2 - acot(b*x + I)/
x - I/(2*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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (60) = 120\).
time = 0.54, size = 498, normalized size = 8.03 \begin {gather*} -\frac {{\left (2 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 2 \, a \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 2 \, a \arctan \left (\frac {1}{b x + a}\right ) - 4 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right )\right )} b}{2 \, {\left (2 \, a^{3} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - a^{2} + 2 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a)))^2 + 2*a*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))*t
an(1/2*arctan(1/(b*x + a))) + 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 - 2*a*a
rctan(1/(b*x + a)) - 4*arctan(1/(b*x + a))*tan(1/2*arctan(1/(b*x + a))) - 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*a^3*tan(1/2*arctan(1/(b*x + a))) + a^2*tan(1/2*arctan(1/(b*x + a)))^2 - a^2 + 2*a*tan(1/2*arctan(1
/(b*x + a))) + tan(1/2*arctan(1/(b*x + a)))^2 - 1)

________________________________________________________________________________________

Mupad [B]
time = 1.22, size = 62, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {acot}\left (a+b\,x\right )}{x}-\frac {b\,x\,\ln \left (x\right )-\frac {b\,x\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2}+a\,b\,x\,\mathrm {acot}\left (a+b\,x\right )}{x\,\left (a^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a + b*x)/x^2,x)

[Out]

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

________________________________________________________________________________________