\(\int \frac {\cot ^{-1}(a x)}{(c+d x^2)^{5/2}} \, dx\) [16]

Optimal result

Integrand size = 16, antiderivative size = 134 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {a}{3 c \left (a^2 c-d\right ) \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {\left (3 a^2 c-2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{3 c^2 \left (a^2 c-d\right )^{3/2}} \] Output:

1/3*a/c/(a^2*c-d)/(d*x^2+c)^(1/2)+1/3*x*arccot(a*x)/c/(d*x^2+c)^(3/2)+2/3* 
x*arccot(a*x)/c^2/(d*x^2+c)^(1/2)-1/3*(3*a^2*c-2*d)*arctanh(a*(d*x^2+c)^(1 
/2)/(a^2*c-d)^(1/2))/c^2/(a^2*c-d)^(3/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.96 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {-\frac {2 a c}{\left (a^2 c-d\right ) \sqrt {c+d x^2}}-\frac {2 x \left (3 c+2 d x^2\right ) \cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}}+\frac {\left (3 a^2 c-2 d\right ) \log \left (\frac {12 a c^2 \sqrt {a^2 c-d} \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (3 a^2 c-2 d\right ) (i+a x)}\right )}{\left (a^2 c-d\right )^{3/2}}+\frac {\left (3 a^2 c-2 d\right ) \log \left (\frac {12 a c^2 \sqrt {a^2 c-d} \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (3 a^2 c-2 d\right ) (-i+a x)}\right )}{\left (a^2 c-d\right )^{3/2}}}{6 c^2} \] Input:

Integrate[ArcCot[a*x]/(c + d*x^2)^(5/2),x]
 

Output:

-1/6*((-2*a*c)/((a^2*c - d)*Sqrt[c + d*x^2]) - (2*x*(3*c + 2*d*x^2)*ArcCot 
[a*x])/(c + d*x^2)^(3/2) + ((3*a^2*c - 2*d)*Log[(12*a*c^2*Sqrt[a^2*c - d]* 
(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((3*a^2*c - 2*d)*(I + a*x 
))])/(a^2*c - d)^(3/2) + ((3*a^2*c - 2*d)*Log[(12*a*c^2*Sqrt[a^2*c - d]*(a 
*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((3*a^2*c - 2*d)*(-I + a*x) 
)])/(a^2*c - d)^(3/2))/c^2
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5448, 27, 435, 87, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5448

\(\displaystyle a \int \frac {x \left (2 d x^2+3 c\right )}{3 c^2 \left (a^2 x^2+1\right ) \left (d x^2+c\right )^{3/2}}dx+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {x \left (2 d x^2+3 c\right )}{\left (a^2 x^2+1\right ) \left (d x^2+c\right )^{3/2}}dx}{3 c^2}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 435

\(\displaystyle \frac {a \int \frac {2 d x^2+3 c}{\left (a^2 x^2+1\right ) \left (d x^2+c\right )^{3/2}}dx^2}{6 c^2}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {a \left (\frac {\left (3 a^2 c-2 d\right ) \int \frac {1}{\left (a^2 x^2+1\right ) \sqrt {d x^2+c}}dx^2}{a^2 c-d}+\frac {2 c}{\left (a^2 c-d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {a \left (\frac {2 \left (3 a^2 c-2 d\right ) \int \frac {1}{\frac {a^2 x^4}{d}-\frac {a^2 c}{d}+1}d\sqrt {d x^2+c}}{d \left (a^2 c-d\right )}+\frac {2 c}{\left (a^2 c-d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {a \left (\frac {2 c}{\left (a^2 c-d\right ) \sqrt {c+d x^2}}-\frac {2 \left (3 a^2 c-2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{a \left (a^2 c-d\right )^{3/2}}\right )}{6 c^2}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\)

Input:

Int[ArcCot[a*x]/(c + d*x^2)^(5/2),x]
 

Output:

(x*ArcCot[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCot[a*x])/(3*c^2*Sqrt[c 
+ d*x^2]) + (a*((2*c)/((a^2*c - d)*Sqrt[c + d*x^2]) - (2*(3*a^2*c - 2*d)*A 
rcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(a*(a^2*c - d)^(3/2))))/(6*c^ 
2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 221

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

rule 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

rule 5448

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

Maple [F]

\[\int \frac {\operatorname {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

Input:

int(arccot(a*x)/(d*x^2+c)^(5/2),x)
 

Output:

int(arccot(a*x)/(d*x^2+c)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (114) = 228\).

Time = 0.21 (sec) , antiderivative size = 712, normalized size of antiderivative = 5.31 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {{\left (3 \, a^{2} c^{3} + {\left (3 \, a^{2} c d^{2} - 2 \, d^{3}\right )} x^{4} - 2 \, c^{2} d + 2 \, {\left (3 \, a^{2} c^{2} d - 2 \, c d^{2}\right )} x^{2}\right )} \sqrt {a^{2} c - d} \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^{3} c^{3} - a c^{2} d + {\left (a^{3} c^{2} d - a c d^{2}\right )} x^{2} + {\left (2 \, {\left (a^{4} c^{2} d - 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{4} c^{3} - 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \operatorname {arccot}\left (a x\right )\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{4} c^{6} - 2 \, a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{4} c^{4} d^{2} - 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{4} c^{5} d - 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a^{2} c^{3} + {\left (3 \, a^{2} c d^{2} - 2 \, d^{3}\right )} x^{4} - 2 \, c^{2} d + 2 \, {\left (3 \, a^{2} c^{2} d - 2 \, c d^{2}\right )} x^{2}\right )} \sqrt {-a^{2} c + d} \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^{3} c^{3} - a c^{2} d + {\left (a^{3} c^{2} d - a c d^{2}\right )} x^{2} + {\left (2 \, {\left (a^{4} c^{2} d - 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{4} c^{3} - 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \operatorname {arccot}\left (a x\right )\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a^{4} c^{6} - 2 \, a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{4} c^{4} d^{2} - 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{4} c^{5} d - 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \] Input:

integrate(arccot(a*x)/(d*x^2+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*((3*a^2*c^3 + (3*a^2*c*d^2 - 2*d^3)*x^4 - 2*c^2*d + 2*(3*a^2*c^2*d - 
 2*c*d^2)*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*(a^3*c^3 - a*c^2 
*d + (a^3*c^2*d - a*c*d^2)*x^2 + (2*(a^4*c^2*d - 2*a^2*c*d^2 + d^3)*x^3 + 
3*(a^4*c^3 - 2*a^2*c^2*d + c*d^2)*x)*arccot(a*x))*sqrt(d*x^2 + c))/(a^4*c^ 
6 - 2*a^2*c^5*d + c^4*d^2 + (a^4*c^4*d^2 - 2*a^2*c^3*d^3 + c^2*d^4)*x^4 + 
2*(a^4*c^5*d - 2*a^2*c^4*d^2 + c^3*d^3)*x^2), -1/6*((3*a^2*c^3 + (3*a^2*c* 
d^2 - 2*d^3)*x^4 - 2*c^2*d + 2*(3*a^2*c^2*d - 2*c*d^2)*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*(a^3*c^3 - a*c^2*d + (a^3*c 
^2*d - a*c*d^2)*x^2 + (2*(a^4*c^2*d - 2*a^2*c*d^2 + d^3)*x^3 + 3*(a^4*c^3 
- 2*a^2*c^2*d + c*d^2)*x)*arccot(a*x))*sqrt(d*x^2 + c))/(a^4*c^6 - 2*a^2*c 
^5*d + c^4*d^2 + (a^4*c^4*d^2 - 2*a^2*c^3*d^3 + c^2*d^4)*x^4 + 2*(a^4*c^5* 
d - 2*a^2*c^4*d^2 + c^3*d^3)*x^2)]
 

Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(acot(a*x)/(d*x**2+c)**(5/2),x)
 

Output:

Integral(acot(a*x)/(c + d*x**2)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(arccot(a*x)/(d*x^2+c)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(d-a^2*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c - 2 \, d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{2} c^{3} - c^{2} d\right )} \sqrt {-a^{2} c + d} a} + \frac {1}{{\left (a^{2} c^{2} - c d\right )} \sqrt {d x^{2} + c}}\right )} + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \arctan \left (\frac {1}{a x}\right )}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \] Input:

integrate(arccot(a*x)/(d*x^2+c)^(5/2),x, algorithm="giac")
 

Output:

1/3*a*((3*a^2*c - 2*d)*arctan(sqrt(d*x^2 + c)*a/sqrt(-a^2*c + d))/((a^2*c^ 
3 - c^2*d)*sqrt(-a^2*c + d)*a) + 1/((a^2*c^2 - c*d)*sqrt(d*x^2 + c))) + 1/ 
3*x*(2*d*x^2/c^2 + 3/c)*arctan(1/(a*x))/(d*x^2 + c)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:

int(acot(a*x)/(c + d*x^2)^(5/2),x)
 

Output:

int(acot(a*x)/(c + d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathit {acot} \left (a x \right )}{\sqrt {d \,x^{2}+c}\, c^{2}+2 \sqrt {d \,x^{2}+c}\, c d \,x^{2}+\sqrt {d \,x^{2}+c}\, d^{2} x^{4}}d x \] Input:

int(acot(a*x)/(d*x^2+c)^(5/2),x)
 

Output:

int(acot(a*x)/(sqrt(c + d*x**2)*c**2 + 2*sqrt(c + d*x**2)*c*d*x**2 + sqrt( 
c + d*x**2)*d**2*x**4),x)