3.64 \(\int \frac{\cot ^{-1}(a x)}{(c+d x^2)^{9/2}} \, dx\)

Optimal. Leaf size=293 \[ \frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}-\frac{\left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}+\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

[Out]

a/(35*c*(a^2*c - d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c - 6*d))/(105*c^2*(a^2*c - d)^2*(c + d*x^2)^(3/2)) + (a*(
19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c - d)^3*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(7*c*(c + d*x^2)^(7
/2)) + (6*x*ArcCot[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCot[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*Arc
Cot[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*ArcTanh[(a*Sqrt[c +
d*x^2])/Sqrt[a^2*c - d]])/(35*c^4*(a^2*c - d)^(7/2))

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Rubi [A]  time = 1.1551, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {192, 191, 4913, 6688, 12, 6715, 1619, 63, 208} \[ \frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}-\frac{\left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}+\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/(c + d*x^2)^(9/2),x]

[Out]

a/(35*c*(a^2*c - d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c - 6*d))/(105*c^2*(a^2*c - d)^2*(c + d*x^2)^(3/2)) + (a*(
19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c - d)^3*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(7*c*(c + d*x^2)^(7
/2)) + (6*x*ArcCot[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCot[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*Arc
Cot[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*ArcTanh[(a*Sqrt[c +
d*x^2])/Sqrt[a^2*c - d]])/(35*c^4*(a^2*c - d)^(7/2))

Rule 192

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

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 12

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

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 1619

Int[((Px_)*((c_.) + (d_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[c + d*x],
(Px*(c + d*x)^(n + 1/2))/(a + b*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[n + 1/2, 0] &
& GtQ[Expon[Px, x], 2]

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 )^{9/2}} \, dx &=\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+a \int \frac{\frac{x}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x}{35 c^4 \sqrt{c+d x^2}}}{1+a^2 x^2} \, dx\\ &=\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+a \int \frac{x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{35 c^4 \left (1+a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx\\ &=\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{a \int \frac{x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\left (1+a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4}\\ &=\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{a \operatorname{Subst}\left (\int \frac{35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\left (1+a^2 x\right ) (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4}\\ &=\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{a \operatorname{Subst}\left (\int \left (-\frac{5 c^3 d}{\left (a^2 c-d\right ) (c+d x)^{7/2}}-\frac{c^2 \left (11 a^2 c-6 d\right ) d}{\left (-a^2 c+d\right )^2 (c+d x)^{5/2}}+\frac{c d \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{\left (-a^2 c+d\right )^3 (c+d x)^{3/2}}+\frac{35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3}{\left (a^2 c-d\right )^3 \left (1+a^2 x\right ) \sqrt{c+d x}}\right ) \, dx,x,x^2\right )}{70 c^4}\\ &=\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}+\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{\left (a \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+a^2 x\right ) \sqrt{c+d x}} \, dx,x,x^2\right )}{70 c^4 \left (a^2 c-d\right )^3}\\ &=\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}+\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{\left (a \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right )\right ) \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 )}{35 c^4 \left (a^2 c-d\right )^3 d}\\ &=\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}+\frac{x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}\\ \end{align*}

Mathematica [C]  time = 1.39481, size = 450, normalized size = 1.54 \[ \frac{\frac{2 a c \left (3 \left (19 a^4 c^2-22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2+3 c^2 \left (d-a^2 c\right )^2+c \left (11 a^2 c-6 d\right ) \left (a^2 c-d\right ) \left (c+d x^2\right )\right )}{\left (a^2 c-d\right )^3 \left (c+d x^2\right )^{5/2}}-\frac{3 \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \log \left (\frac{140 a c^4 \left (a^2 c-d\right )^{5/2} \left (\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right )}{(a x+i) \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right )}\right )}{\left (a^2 c-d\right )^{7/2}}-\frac{3 \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \log \left (\frac{140 a c^4 \left (a^2 c-d\right )^{5/2} \left (\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right )}{(a x-i) \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right )}\right )}{\left (a^2 c-d\right )^{7/2}}+\frac{6 x \cot ^{-1}(a x) \left (70 c^2 d x^2+35 c^3+56 c d^2 x^4+16 d^3 x^6\right )}{\left (c+d x^2\right )^{7/2}}}{210 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]/(c + d*x^2)^(9/2),x]

[Out]

((2*a*c*(3*c^2*(-(a^2*c) + d)^2 + c*(11*a^2*c - 6*d)*(a^2*c - d)*(c + d*x^2) + 3*(19*a^4*c^2 - 22*a^2*c*d + 8*
d^2)*(c + d*x^2)^2))/((a^2*c - d)^3*(c + d*x^2)^(5/2)) + (6*x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x
^6)*ArcCot[a*x])/(c + d*x^2)^(7/2) - (3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*Log[(140*a*c^4*(a^
2*c - d)^(5/2)*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 1
6*d^3)*(I + a*x))])/(a^2*c - d)^(7/2) - (3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*Log[(140*a*c^4*
(a^2*c - d)^(5/2)*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2
- 16*d^3)*(-I + a*x))])/(a^2*c - d)^(7/2))/(210*c^4)

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Maple [F]  time = 0.725, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccot} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)/(d*x^2+c)^(9/2),x)

[Out]

int(arccot(a*x)/(d*x^2+c)^(9/2),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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.70193, size = 4097, normalized size = 13.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/(d*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

[1/420*(3*(35*a^6*c^7 - 70*a^4*c^6*d + 56*a^2*c^5*d^2 + (35*a^6*c^3*d^4 - 70*a^4*c^2*d^5 + 56*a^2*c*d^6 - 16*d
^7)*x^8 - 16*c^4*d^3 + 4*(35*a^6*c^4*d^3 - 70*a^4*c^3*d^4 + 56*a^2*c^2*d^5 - 16*c*d^6)*x^6 + 6*(35*a^6*c^5*d^2
- 70*a^4*c^4*d^3 + 56*a^2*c^3*d^4 - 16*c^2*d^5)*x^4 + 4*(35*a^6*c^6*d - 70*a^4*c^5*d^2 + 56*a^2*c^4*d^3 - 16*
c^3*d^4)*x^2)*sqrt(a^2*c - d)*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)) + 4*(71*a^7*c^7 - 1
60*a^5*c^6*d + 122*a^3*c^5*d^2 - 33*a*c^4*d^3 + 3*(19*a^7*c^4*d^3 - 41*a^5*c^3*d^4 + 30*a^3*c^2*d^5 - 8*a*c*d^
6)*x^6 + (182*a^7*c^5*d^2 - 397*a^5*c^4*d^3 + 293*a^3*c^3*d^4 - 78*a*c^2*d^5)*x^4 + (196*a^7*c^6*d - 434*a^5*c
^5*d^2 + 325*a^3*c^4*d^3 - 87*a*c^3*d^4)*x^2 + 3*(16*(a^8*c^4*d^3 - 4*a^6*c^3*d^4 + 6*a^4*c^2*d^5 - 4*a^2*c*d^
6 + d^7)*x^7 + 56*(a^8*c^5*d^2 - 4*a^6*c^4*d^3 + 6*a^4*c^3*d^4 - 4*a^2*c^2*d^5 + c*d^6)*x^5 + 70*(a^8*c^6*d -
4*a^6*c^5*d^2 + 6*a^4*c^4*d^3 - 4*a^2*c^3*d^4 + c^2*d^5)*x^3 + 35*(a^8*c^7 - 4*a^6*c^6*d + 6*a^4*c^5*d^2 - 4*a
^2*c^4*d^3 + c^3*d^4)*x)*arccot(a*x))*sqrt(d*x^2 + c))/(a^8*c^12 - 4*a^6*c^11*d + 6*a^4*c^10*d^2 - 4*a^2*c^9*d
^3 + c^8*d^4 + (a^8*c^8*d^4 - 4*a^6*c^7*d^5 + 6*a^4*c^6*d^6 - 4*a^2*c^5*d^7 + c^4*d^8)*x^8 + 4*(a^8*c^9*d^3 -
4*a^6*c^8*d^4 + 6*a^4*c^7*d^5 - 4*a^2*c^6*d^6 + c^5*d^7)*x^6 + 6*(a^8*c^10*d^2 - 4*a^6*c^9*d^3 + 6*a^4*c^8*d^4
- 4*a^2*c^7*d^5 + c^6*d^6)*x^4 + 4*(a^8*c^11*d - 4*a^6*c^10*d^2 + 6*a^4*c^9*d^3 - 4*a^2*c^8*d^4 + c^7*d^5)*x^
2), -1/210*(3*(35*a^6*c^7 - 70*a^4*c^6*d + 56*a^2*c^5*d^2 + (35*a^6*c^3*d^4 - 70*a^4*c^2*d^5 + 56*a^2*c*d^6 -
16*d^7)*x^8 - 16*c^4*d^3 + 4*(35*a^6*c^4*d^3 - 70*a^4*c^3*d^4 + 56*a^2*c^2*d^5 - 16*c*d^6)*x^6 + 6*(35*a^6*c^5
*d^2 - 70*a^4*c^4*d^3 + 56*a^2*c^3*d^4 - 16*c^2*d^5)*x^4 + 4*(35*a^6*c^6*d - 70*a^4*c^5*d^2 + 56*a^2*c^4*d^3 -
16*c^3*d^4)*x^2)*sqrt(-a^2*c + d)*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)) - 2*(71*a^7*c^7 - 160*a^5*c^6*d + 122*a^3*c^5*d^2 - 33*a*c^4*d^3 + 3*(1
9*a^7*c^4*d^3 - 41*a^5*c^3*d^4 + 30*a^3*c^2*d^5 - 8*a*c*d^6)*x^6 + (182*a^7*c^5*d^2 - 397*a^5*c^4*d^3 + 293*a^
3*c^3*d^4 - 78*a*c^2*d^5)*x^4 + (196*a^7*c^6*d - 434*a^5*c^5*d^2 + 325*a^3*c^4*d^3 - 87*a*c^3*d^4)*x^2 + 3*(16
*(a^8*c^4*d^3 - 4*a^6*c^3*d^4 + 6*a^4*c^2*d^5 - 4*a^2*c*d^6 + d^7)*x^7 + 56*(a^8*c^5*d^2 - 4*a^6*c^4*d^3 + 6*a
^4*c^3*d^4 - 4*a^2*c^2*d^5 + c*d^6)*x^5 + 70*(a^8*c^6*d - 4*a^6*c^5*d^2 + 6*a^4*c^4*d^3 - 4*a^2*c^3*d^4 + c^2*
d^5)*x^3 + 35*(a^8*c^7 - 4*a^6*c^6*d + 6*a^4*c^5*d^2 - 4*a^2*c^4*d^3 + c^3*d^4)*x)*arccot(a*x))*sqrt(d*x^2 + c
))/(a^8*c^12 - 4*a^6*c^11*d + 6*a^4*c^10*d^2 - 4*a^2*c^9*d^3 + c^8*d^4 + (a^8*c^8*d^4 - 4*a^6*c^7*d^5 + 6*a^4*
c^6*d^6 - 4*a^2*c^5*d^7 + c^4*d^8)*x^8 + 4*(a^8*c^9*d^3 - 4*a^6*c^8*d^4 + 6*a^4*c^7*d^5 - 4*a^2*c^6*d^6 + c^5*
d^7)*x^6 + 6*(a^8*c^10*d^2 - 4*a^6*c^9*d^3 + 6*a^4*c^8*d^4 - 4*a^2*c^7*d^5 + c^6*d^6)*x^4 + 4*(a^8*c^11*d - 4*
a^6*c^10*d^2 + 6*a^4*c^9*d^3 - 4*a^2*c^8*d^4 + c^7*d^5)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/(d*x**2+c)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23049, size = 459, normalized size = 1.57 \begin{align*} \frac{1}{105} \, a{\left (\frac{3 \,{\left (35 \, a^{6} c^{3} - 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} - 16 \, d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} a}{\sqrt{-a^{2} c + d}}\right )}{{\left (a^{6} c^{7} - 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} - c^{4} d^{3}\right )} \sqrt{-a^{2} c + d} a} + \frac{57 \,{\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \,{\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} - 66 \,{\left (d x^{2} + c\right )}^{2} a^{2} c d - 17 \,{\left (d x^{2} + c\right )} a^{2} c^{2} d - 6 \, a^{2} c^{3} d + 24 \,{\left (d x^{2} + c\right )}^{2} d^{2} + 6 \,{\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} - 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} - c^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\right )} + \frac{{\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \, d^{3} x^{2}}{c^{4}} + \frac{7 \, d^{2}}{c^{3}}\right )} + \frac{35 \, d}{c^{2}}\right )} x^{2} + \frac{35}{c}\right )} x \arctan \left (\frac{1}{a x}\right )}{35 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/(d*x^2+c)^(9/2),x, algorithm="giac")

[Out]

1/105*a*(3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*arctan(sqrt(d*x^2 + c)*a/sqrt(-a^2*c + d))/((a^
6*c^7 - 3*a^4*c^6*d + 3*a^2*c^5*d^2 - c^4*d^3)*sqrt(-a^2*c + d)*a) + (57*(d*x^2 + c)^2*a^4*c^2 + 11*(d*x^2 + c
)*a^4*c^3 + 3*a^4*c^4 - 66*(d*x^2 + c)^2*a^2*c*d - 17*(d*x^2 + c)*a^2*c^2*d - 6*a^2*c^3*d + 24*(d*x^2 + c)^2*d
^2 + 6*(d*x^2 + c)*c*d^2 + 3*c^2*d^2)/((a^6*c^6 - 3*a^4*c^5*d + 3*a^2*c^4*d^2 - c^3*d^3)*(d*x^2 + c)^(5/2))) +
1/35*(2*(4*x^2*(2*d^3*x^2/c^4 + 7*d^2/c^3) + 35*d/c^2)*x^2 + 35/c)*x*arctan(1/(a*x))/(d*x^2 + c)^(7/2)