3.132 \(\int (a+b \cot ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=38 \[ a x+\frac{b \log \left ((c+d x)^2+1\right )}{2 d}+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d} \]

[Out]

a*x + (b*(c + d*x)*ArcCot[c + d*x])/d + (b*Log[1 + (c + d*x)^2])/(2*d)

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Rubi [A]  time = 0.0201107, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5040, 4847, 260} \[ a x+\frac{b \log \left ((c+d x)^2+1\right )}{2 d}+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCot[c + d*x],x]

[Out]

a*x + (b*(c + d*x)*ArcCot[c + d*x])/d + (b*Log[1 + (c + d*x)^2])/(2*d)

Rule 5040

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCot[x])^p, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]

Rule 4847

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x])^p, x] + Dist[b*c*p, Int[
(x*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 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 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx &=a x+b \int \cot ^{-1}(c+d x) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \cot ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d}+\frac{b \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=a x+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d}+\frac{b \log \left (1+(c+d x)^2\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0120605, size = 49, normalized size = 1.29 \[ a x+\frac{b \left (\log \left (c^2+2 c d x+d^2 x^2+1\right )-2 c \tan ^{-1}(c+d x)\right )}{2 d}+b x \cot ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCot[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.041, size = 42, normalized size = 1.1 \begin{align*} ax+b{\rm arccot} \left (dx+c\right )x+{\frac{b{\rm arccot} \left (dx+c\right )c}{d}}+{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arccot(d*x+c),x)

[Out]

a*x+b*arccot(d*x+c)*x+b/d*arccot(d*x+c)*c+1/2*b*ln(1+(d*x+c)^2)/d

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Maxima [A]  time = 0.985581, size = 46, normalized size = 1.21 \begin{align*} a x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccot(d*x+c),x, algorithm="maxima")

[Out]

a*x + 1/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*b/d

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Fricas [A]  time = 2.14706, size = 140, normalized size = 3.68 \begin{align*} \frac{2 \, b d x \operatorname{arccot}\left (d x + c\right ) + 2 \, a d x - 2 \, b c \arctan \left (d x + c\right ) + b \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccot(d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.490215, size = 51, normalized size = 1.34 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{acot}{\left (c + d x \right )}}{d} + x \operatorname{acot}{\left (c + d x \right )} + \frac{\log{\left (c^{2} + 2 c d x + d^{2} x^{2} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\x \operatorname{acot}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*acot(d*x+c),x)

[Out]

a*x + b*Piecewise((c*acot(c + d*x)/d + x*acot(c + d*x) + log(c**2 + 2*c*d*x + d**2*x**2 + 1)/(2*d), Ne(d, 0)),
(x*acot(c), True))

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Giac [A]  time = 1.09487, size = 77, normalized size = 2.03 \begin{align*} -\frac{1}{2} \,{\left (d{\left (\frac{2 \, c \arctan \left (d x + c\right )}{d^{2}} - \frac{\log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{2}}\right )} - 2 \, x \arctan \left (\frac{1}{d x + c}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccot(d*x+c),x, algorithm="giac")

[Out]

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