3.61 \(\int \frac{\cot ^{-1}(a x)}{(c+d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{c \sqrt{a^2 c-d}} \]

[Out]

(x*ArcCot[a*x])/(c*Sqrt[c + d*x^2]) - ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]]/(c*Sqrt[a^2*c - d])

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Rubi [A]  time = 0.0940461, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {191, 4913, 12, 444, 63, 208} \[ \frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{c \sqrt{a^2 c-d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/(c + d*x^2)^(3/2),x]

[Out]

(x*ArcCot[a*x])/(c*Sqrt[c + d*x^2]) - ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]]/(c*Sqrt[a^2*c - d])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 4913

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}+a \int \frac{x}{c \left (1+a^2 x^2\right ) \sqrt{c+d x^2}} \, dx\\ &=\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}+\frac{a \int \frac{x}{\left (1+a^2 x^2\right ) \sqrt{c+d x^2}} \, dx}{c}\\ &=\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (1+a^2 x\right ) \sqrt{c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-\frac{a^2 c}{d}+\frac{a^2 x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{c d}\\ &=\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{c \sqrt{a^2 c-d}}\\ \end{align*}

Mathematica [C]  time = 0.227145, size = 169, normalized size = 2.56 \[ \frac{\frac{2 x \cot ^{-1}(a x)}{\sqrt{c+d x^2}}+\frac{-\log \left (\frac{4 a c \left (\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right )}{(a x+i) \sqrt{a^2 c-d}}\right )-\log \left (\frac{4 a c \left (\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right )}{(a x-i) \sqrt{a^2 c-d}}\right )}{\sqrt{a^2 c-d}}}{2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]/(c + d*x^2)^(3/2),x]

[Out]

((2*x*ArcCot[a*x])/Sqrt[c + d*x^2] + (-Log[(4*a*c*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/(Sqrt[a^2*c
- d]*(I + a*x))] - Log[(4*a*c*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/(Sqrt[a^2*c - d]*(-I + a*x))])
/Sqrt[a^2*c - d])/(2*c)

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Maple [F]  time = 0.644, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccot} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)/(d*x^2+c)^(3/2),x)

[Out]

int(arccot(a*x)/(d*x^2+c)^(3/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.32864, size = 729, normalized size = 11.05 \begin{align*} \left [\frac{4 \,{\left (a^{2} c - d\right )} \sqrt{d x^{2} + c} x \operatorname{arccot}\left (a x\right ) + \sqrt{a^{2} c - d}{\left (d x^{2} + c\right )} \log \left (\frac{a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} - 8 \, a^{2} c d + 2 \,{\left (4 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} - 4 \,{\left (a^{3} d x^{2} + 2 \, a^{3} c - a d\right )} \sqrt{a^{2} c - d} \sqrt{d x^{2} + c} + d^{2}}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right )}{4 \,{\left (a^{2} c^{3} - c^{2} d +{\left (a^{2} c^{2} d - c d^{2}\right )} x^{2}\right )}}, \frac{2 \,{\left (a^{2} c - d\right )} \sqrt{d x^{2} + c} x \operatorname{arccot}\left (a x\right ) - \sqrt{-a^{2} c + d}{\left (d x^{2} + c\right )} \arctan \left (-\frac{{\left (a^{2} d x^{2} + 2 \, a^{2} c - d\right )} \sqrt{-a^{2} c + d} \sqrt{d x^{2} + c}}{2 \,{\left (a^{3} c^{2} - a c d +{\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \,{\left (a^{2} c^{3} - c^{2} d +{\left (a^{2} c^{2} d - c d^{2}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(4*(a^2*c - d)*sqrt(d*x^2 + c)*x*arccot(a*x) + sqrt(a^2*c - d)*(d*x^2 + c)*log((a^4*d^2*x^4 + 8*a^4*c^2 -
8*a^2*c*d + 2*(4*a^4*c*d - 3*a^2*d^2)*x^2 - 4*(a^3*d*x^2 + 2*a^3*c - a*d)*sqrt(a^2*c - d)*sqrt(d*x^2 + c) + d
^2)/(a^4*x^4 + 2*a^2*x^2 + 1)))/(a^2*c^3 - c^2*d + (a^2*c^2*d - c*d^2)*x^2), 1/2*(2*(a^2*c - d)*sqrt(d*x^2 + c
)*x*arccot(a*x) - sqrt(-a^2*c + d)*(d*x^2 + c)*arctan(-1/2*(a^2*d*x^2 + 2*a^2*c - d)*sqrt(-a^2*c + d)*sqrt(d*x
^2 + c)/(a^3*c^2 - a*c*d + (a^3*c*d - a*d^2)*x^2)))/(a^2*c^3 - c^2*d + (a^2*c^2*d - c*d^2)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/(d*x**2+c)**(3/2),x)

[Out]

Integral(acot(a*x)/(c + d*x**2)**(3/2), x)

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Giac [A]  time = 1.14364, size = 80, normalized size = 1.21 \begin{align*} \frac{x \arctan \left (\frac{1}{a x}\right )}{\sqrt{d x^{2} + c} c} + \frac{\arctan \left (\frac{\sqrt{d x^{2} + c} a}{\sqrt{-a^{2} c + d}}\right )}{\sqrt{-a^{2} c + d} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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