Integrand size = 16, antiderivative size = 293 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}} \]
1/35*a/c/(a^2*c-d)/(d*x^2+c)^(5/2)+1/105*a*(11*a^2*c-6*d)/c^2/(a^2*c-d)^2/ (d*x^2+c)^(3/2)+1/7*x*arccot(a*x)/c/(d*x^2+c)^(7/2)+6/35*x*arccot(a*x)/c^2 /(d*x^2+c)^(5/2)+8/35*x*arccot(a*x)/c^3/(d*x^2+c)^(3/2)-1/35*(35*a^6*c^3-7 0*a^4*c^2*d+56*a^2*c*d^2-16*d^3)*arctanh(a*(d*x^2+c)^(1/2)/(a^2*c-d)^(1/2) )/c^4/(a^2*c-d)^(7/2)+1/35*a*(19*a^4*c^2-22*a^2*c*d+8*d^2)/c^3/(a^2*c-d)^3 /(d*x^2+c)^(1/2)+16/35*x*arccot(a*x)/c^4/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\frac {2 a c \left (3 c^2 \left (-a^2 c+d\right )^2+c \left (11 a^2 c-6 d\right ) \left (a^2 c-d\right ) \left (c+d x^2\right )+3 \left (19 a^4 c^2-22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2\right )}{\left (a^2 c-d\right )^3 \left (c+d x^2\right )^{5/2}}+\frac {6 x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right ) \cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}}-\frac {3 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \log \left (\frac {140 a c^4 \left (a^2 c-d\right )^{5/2} \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) (i+a x)}\right )}{\left (a^2 c-d\right )^{7/2}}-\frac {3 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \log \left (\frac {140 a c^4 \left (a^2 c-d\right )^{5/2} \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) (-i+a x)}\right )}{\left (a^2 c-d\right )^{7/2}}}{210 c^4} \]
((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 - 16*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)
Time = 1.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5448, 27, 7266, 2122, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 5448 |
| \(\displaystyle a \int \frac {x \left (16 d^3 x^6+56 c d^2 x^4+70 c^2 d x^2+35 c^3\right )}{35 c^4 \left (a^2 x^2+1\right ) \left (d x^2+c\right )^{7/2}}dx+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {x \left (16 d^3 x^6+56 c d^2 x^4+70 c^2 d x^2+35 c^3\right )}{\left (a^2 x^2+1\right ) \left (d x^2+c\right )^{7/2}}dx}{35 c^4}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {a \int \frac {16 d^3 x^6+56 c d^2 x^4+70 c^2 d x^2+35 c^3}{\left (a^2 x^2+1\right ) \left (d x^2+c\right )^{7/2}}dx^2}{70 c^4}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2122 |
| \(\displaystyle \frac {a \int \left (-\frac {5 d c^3}{\left (a^2 c-d\right ) \left (d x^2+c\right )^{7/2}}-\frac {\left (11 a^2 c-6 d\right ) d c^2}{\left (d-a^2 c\right )^2 \left (d x^2+c\right )^{5/2}}+\frac {d \left (19 c^2 a^4-22 c d a^2+8 d^2\right ) c}{\left (d-a^2 c\right )^3 \left (d x^2+c\right )^{3/2}}+\frac {35 c^3 a^6-70 c^2 d a^4+56 c d^2 a^2-16 d^3}{\left (a^2 c-d\right )^3 \left (a^2 x^2+1\right ) \sqrt {d x^2+c}}\right )dx^2}{70 c^4}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (\frac {2 c^3}{\left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac {2 c^2 \left (11 a^2 c-6 d\right )}{3 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {2 c \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{\left (a^2 c-d\right )^3 \sqrt {c+d x^2}}-\frac {2 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{a \left (a^2 c-d\right )^{7/2}}\right )}{70 c^4}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}\) |
(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*ArcCot [a*x])/(35*c^4*Sqrt[c + d*x^2]) + (a*((2*c^3)/((a^2*c - d)*(c + d*x^2)^(5/ 2)) + (2*c^2*(11*a^2*c - 6*d))/(3*(a^2*c - d)^2*(c + d*x^2)^(3/2)) + (2*c* (19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/((a^2*c - d)^3*Sqrt[c + d*x^2]) - (2*(3 5*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]])/(a*(a^2*c - d)^(7/2))))/(70*c^4)
3.1.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((Px_)*((c_.) + (d_.)*(x_))^(n_))/((a_.) + (b_.)*(x_)), x_Symbol] :> In t[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]
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])
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]
\[\int \frac {\operatorname {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (257) = 514\).
Time = 0.61 (sec) , antiderivative size = 1986, normalized size of antiderivative = 6.78 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
[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 - 160*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...
\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {9}{2}}}\, dx \]
Exception generated. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {1}{105} \, a {\left (\frac {3 \, {\left (35 \, a^{6} c^{3} - 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} - 16 \, d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{6} c^{7} - 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} - c^{4} d^{3}\right )} \sqrt {-a^{2} c + d} a} + \frac {57 \, {\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \, {\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} - 66 \, {\left (d x^{2} + c\right )}^{2} a^{2} c d - 17 \, {\left (d x^{2} + c\right )} a^{2} c^{2} d - 6 \, a^{2} c^{3} d + 24 \, {\left (d x^{2} + c\right )}^{2} d^{2} + 6 \, {\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} - 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} - c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \arctan \left (\frac {1}{a x}\right )}{35 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]
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))) + 1/3 5*(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)
Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]