\(\int \frac {\cot ^{-1}(a x)}{(c+d x^2)^{7/2}} \, dx\) [17]

Optimal result

Integrand size = 16, antiderivative size = 208 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {a}{15 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c-4 d\right )}{15 c^2 \left (a^2 c-d\right )^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{15 c^3 \left (a^2 c-d\right )^{5/2}} \] Output:

1/15*a/c/(a^2*c-d)/(d*x^2+c)^(3/2)+1/15*a*(7*a^2*c-4*d)/c^2/(a^2*c-d)^2/(d 
*x^2+c)^(1/2)+1/5*x*arccot(a*x)/c/(d*x^2+c)^(5/2)+4/15*x*arccot(a*x)/c^2/( 
d*x^2+c)^(3/2)+8/15*x*arccot(a*x)/c^3/(d*x^2+c)^(1/2)-1/15*(15*a^4*c^2-20* 
a^2*c*d+8*d^2)*arctanh(a*(d*x^2+c)^(1/2)/(a^2*c-d)^(1/2))/c^3/(a^2*c-d)^(5 
/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=-\frac {-\frac {2 a c \left (-d \left (5 c+4 d x^2\right )+a^2 c \left (8 c+7 d x^2\right )\right )}{\left (-a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac {2 x \left (15 c^2+20 c d x^2+8 d^2 x^4\right ) \cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}}+\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \log \left (\frac {60 a c^3 \left (a^2 c-d\right )^{3/2} \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) (i+a x)}\right )}{\left (a^2 c-d\right )^{5/2}}+\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \log \left (\frac {60 a c^3 \left (a^2 c-d\right )^{3/2} \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) (-i+a x)}\right )}{\left (a^2 c-d\right )^{5/2}}}{30 c^3} \] Input:

Integrate[ArcCot[a*x]/(c + d*x^2)^(7/2),x]
 

Output:

-1/30*((-2*a*c*(-(d*(5*c + 4*d*x^2)) + a^2*c*(8*c + 7*d*x^2)))/((-(a^2*c) 
+ d)^2*(c + d*x^2)^(3/2)) - (2*x*(15*c^2 + 20*c*d*x^2 + 8*d^2*x^4)*ArcCot[ 
a*x])/(c + d*x^2)^(5/2) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*Log[(60*a*c^3 
*(a^2*c - d)^(3/2)*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((15*a 
^4*c^2 - 20*a^2*c*d + 8*d^2)*(I + a*x))])/(a^2*c - d)^(5/2) + ((15*a^4*c^2 
 - 20*a^2*c*d + 8*d^2)*Log[(60*a*c^3*(a^2*c - d)^(3/2)*(a*c + I*d*x + Sqrt 
[a^2*c - d]*Sqrt[c + d*x^2]))/((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*(-I + a*x 
))])/(a^2*c - d)^(5/2))/c^3
 

Rubi [A] (warning: unable to verify)

Time = 1.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5448, 27, 7266, 1192, 25, 1584, 2009}

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.

\(\displaystyle \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5448

\(\displaystyle a \int \frac {x \left (8 d^2 x^4+20 c d x^2+15 c^2\right )}{15 c^3 \left (a^2 x^2+1\right ) \left (d x^2+c\right )^{5/2}}dx+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {x \left (8 d^2 x^4+20 c d x^2+15 c^2\right )}{\left (a^2 x^2+1\right ) \left (d x^2+c\right )^{5/2}}dx}{15 c^3}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {a \int \frac {8 d^2 x^4+20 c d x^2+15 c^2}{\left (a^2 x^2+1\right ) \left (d x^2+c\right )^{5/2}}dx^2}{30 c^3}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1192

\(\displaystyle \frac {a \int -\frac {8 d^2 x^8+4 c d^2 x^4+3 c^2 d^2}{x^8 \left (-a^2 x^4+a^2 c-d\right )}d\sqrt {d x^2+c}}{15 c^3 d^2}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a \int \frac {8 d^2 x^8+4 c d^2 x^4+3 c^2 d^2}{x^8 \left (-a^2 x^4+a^2 c-d\right )}d\sqrt {d x^2+c}}{15 c^3 d^2}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1584

\(\displaystyle -\frac {a \int \left (-\frac {\left (15 c^2 a^4-20 c d a^2+8 d^2\right ) d^2}{\left (d-a^2 c\right )^2 \left (a^2 x^4-a^2 c+d\right )}+\frac {c \left (7 a^2 c-4 d\right ) d^2}{\left (a^2 c-d\right )^2 x^4}+\frac {3 c^2 d^2}{\left (a^2 c-d\right ) x^8}\right )d\sqrt {d x^2+c}}{15 c^3 d^2}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \left (\frac {c^2 d^2}{x^6 \left (a^2 c-d\right )}+\frac {c d^2 \left (7 a^2 c-4 d\right )}{x^2 \left (a^2 c-d\right )^2}-\frac {d^2 \left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{a \left (a^2 c-d\right )^{5/2}}\right )}{15 c^3 d^2}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}\)

Input:

Int[ArcCot[a*x]/(c + d*x^2)^(7/2),x]
 

Output:

(x*ArcCot[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCot[a*x])/(15*c^2*(c + d 
*x^2)^(3/2)) + (8*x*ArcCot[a*x])/(15*c^3*Sqrt[c + d*x^2]) + (a*((c^2*d^2)/ 
((a^2*c - d)*x^6) + (c*(7*a^2*c - 4*d)*d^2)/((a^2*c - d)^2*x^2) - (d^2*(15 
*a^4*c^2 - 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d] 
])/(a*(a^2*c - d)^(5/2))))/(15*c^3*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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])
 

rule 7266

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

Maple [F]

\[\int \frac {\operatorname {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}d x\]

Input:

int(arccot(a*x)/(d*x^2+c)^(7/2),x)
 

Output:

int(arccot(a*x)/(d*x^2+c)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (180) = 360\).

Time = 0.23 (sec) , antiderivative size = 1278, normalized size of antiderivative = 6.14 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(arccot(a*x)/(d*x^2+c)^(7/2),x, algorithm="fricas")
 

Output:

[1/60*((15*a^4*c^5 - 20*a^2*c^4*d + (15*a^4*c^2*d^3 - 20*a^2*c*d^4 + 8*d^5 
)*x^6 + 8*c^3*d^2 + 3*(15*a^4*c^3*d^2 - 20*a^2*c^2*d^3 + 8*c*d^4)*x^4 + 3* 
(15*a^4*c^4*d - 20*a^2*c^3*d^2 + 8*c^2*d^3)*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*(8*a^5*c^5 - 13*a^3*c^4*d + 5*a*c^3*d^2 + (7*a^5*c^3*d^2 
 - 11*a^3*c^2*d^3 + 4*a*c*d^4)*x^4 + 3*(5*a^5*c^4*d - 8*a^3*c^3*d^2 + 3*a* 
c^2*d^3)*x^2 + (8*(a^6*c^3*d^2 - 3*a^4*c^2*d^3 + 3*a^2*c*d^4 - d^5)*x^5 + 
20*(a^6*c^4*d - 3*a^4*c^3*d^2 + 3*a^2*c^2*d^3 - c*d^4)*x^3 + 15*(a^6*c^5 - 
 3*a^4*c^4*d + 3*a^2*c^3*d^2 - c^2*d^3)*x)*arccot(a*x))*sqrt(d*x^2 + c))/( 
a^6*c^9 - 3*a^4*c^8*d + 3*a^2*c^7*d^2 - c^6*d^3 + (a^6*c^6*d^3 - 3*a^4*c^5 
*d^4 + 3*a^2*c^4*d^5 - c^3*d^6)*x^6 + 3*(a^6*c^7*d^2 - 3*a^4*c^6*d^3 + 3*a 
^2*c^5*d^4 - c^4*d^5)*x^4 + 3*(a^6*c^8*d - 3*a^4*c^7*d^2 + 3*a^2*c^6*d^3 - 
 c^5*d^4)*x^2), -1/30*((15*a^4*c^5 - 20*a^2*c^4*d + (15*a^4*c^2*d^3 - 20*a 
^2*c*d^4 + 8*d^5)*x^6 + 8*c^3*d^2 + 3*(15*a^4*c^3*d^2 - 20*a^2*c^2*d^3 + 8 
*c*d^4)*x^4 + 3*(15*a^4*c^4*d - 20*a^2*c^3*d^2 + 8*c^2*d^3)*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*(8*a^5*c^5 - 13*a^3*c^ 
4*d + 5*a*c^3*d^2 + (7*a^5*c^3*d^2 - 11*a^3*c^2*d^3 + 4*a*c*d^4)*x^4 + 3*( 
5*a^5*c^4*d - 8*a^3*c^3*d^2 + 3*a*c^2*d^3)*x^2 + (8*(a^6*c^3*d^2 - 3*a^...
 

Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(acot(a*x)/(d*x**2+c)**(7/2),x)
 

Output:

Integral(acot(a*x)/(c + d*x**2)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(arccot(a*x)/(d*x^2+c)^(7/2),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 = 208, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {1}{15} \, a {\left (\frac {{\left (15 \, a^{4} c^{2} - 20 \, a^{2} c d + 8 \, d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{4} c^{5} - 2 \, a^{2} c^{4} d + c^{3} d^{2}\right )} \sqrt {-a^{2} c + d} a} + \frac {7 \, {\left (d x^{2} + c\right )} a^{2} c + a^{2} c^{2} - 4 \, {\left (d x^{2} + c\right )} d - c d}{{\left (a^{4} c^{4} - 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \arctan \left (\frac {1}{a x}\right )}{15 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \] Input:

integrate(arccot(a*x)/(d*x^2+c)^(7/2),x, algorithm="giac")
 

Output:

1/15*a*((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*arctan(sqrt(d*x^2 + c)*a/sqrt(-a 
^2*c + d))/((a^4*c^5 - 2*a^2*c^4*d + c^3*d^2)*sqrt(-a^2*c + d)*a) + (7*(d* 
x^2 + c)*a^2*c + a^2*c^2 - 4*(d*x^2 + c)*d - c*d)/((a^4*c^4 - 2*a^2*c^3*d 
+ c^2*d^2)*(d*x^2 + c)^(3/2))) + 1/15*(4*x^2*(2*d^2*x^2/c^3 + 5*d/c^2) + 1 
5/c)*x*arctan(1/(a*x))/(d*x^2 + c)^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int(acot(a*x)/(c + d*x^2)^(7/2),x)
 

Output:

int(acot(a*x)/(c + d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\mathit {acot} \left (a x \right )}{\sqrt {d \,x^{2}+c}\, c^{3}+3 \sqrt {d \,x^{2}+c}\, c^{2} d \,x^{2}+3 \sqrt {d \,x^{2}+c}\, c \,d^{2} x^{4}+\sqrt {d \,x^{2}+c}\, d^{3} x^{6}}d x \] Input:

int(acot(a*x)/(d*x^2+c)^(7/2),x)
 

Output:

int(acot(a*x)/(sqrt(c + d*x**2)*c**3 + 3*sqrt(c + d*x**2)*c**2*d*x**2 + 3* 
sqrt(c + d*x**2)*c*d**2*x**4 + sqrt(c + d*x**2)*d**3*x**6),x)