Integrand size = 14, antiderivative size = 244 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9} \] Output:
1/630*d*(420*a^6*c^3-378*a^4*c^2*d+180*a^2*c*d^2-35*d^3)*x^2/a^7+1/1260*d^ 2*(378*a^4*c^2-180*a^2*c*d+35*d^2)*x^4/a^5+1/378*(36*a^2*c-7*d)*d^3*x^6/a^ 3+1/72*d^4*x^8/a+c^4*x*arccot(a*x)+4/3*c^3*d*x^3*arccot(a*x)+6/5*c^2*d^2*x ^5*arccot(a*x)+4/7*c*d^3*x^7*arccot(a*x)+1/9*d^4*x^9*arccot(a*x)+1/630*(31 5*a^8*c^4-420*a^6*c^3*d+378*a^4*c^2*d^2-180*a^2*c*d^3+35*d^4)*ln(a^2*x^2+1 )/a^9
Time = 0.05 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.87 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (-420 d^3+30 a^2 d^2 \left (72 c+7 d x^2\right )-4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )\right )+24 a^9 x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \cot ^{-1}(a x)+12 \left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{7560 a^9} \] Input:
Integrate[(c + d*x^2)^4*ArcCot[a*x],x]
Output:
(a^2*d*x^2*(-420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) - 4*a^4*d*(1134*c^2 + 2 70*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(1680*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) + 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 18 0*c*d^3*x^6 + 35*d^4*x^8)*ArcCot[a*x] + 12*(315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(7560*a^9)
Time = 0.67 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5448, 27, 2331, 2389, 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.
| \(\displaystyle \int \cot ^{-1}(a x) \left (c+d x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 5448 |
| \(\displaystyle a \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{315 \left (a^2 x^2+1\right )}dx+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{315} a \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{a^2 x^2+1}dx+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{630} a \int \frac {35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4}{a^2 x^2+1}dx^2+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {1}{630} a \int \left (\frac {35 d^4 x^6}{a^2}+\frac {5 \left (36 a^2 c-7 d\right ) d^3 x^4}{a^4}+\frac {d^2 \left (378 c^2 a^4-180 c d a^2+35 d^2\right ) x^2}{a^6}+\frac {d \left (420 c^3 a^6-378 c^2 d a^4+180 c d^2 a^2-35 d^3\right )}{a^8}+\frac {315 c^4 a^8-420 c^3 d a^6+378 c^2 d^2 a^4-180 c d^3 a^2+35 d^4}{a^8 \left (a^2 x^2+1\right )}\right )dx^2+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{630} a \left (\frac {35 d^4 x^8}{4 a^2}+\frac {5 d^3 x^6 \left (36 a^2 c-7 d\right )}{3 a^4}+\frac {d^2 x^4 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right )}{2 a^6}+\frac {d x^2 \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right )}{a^8}+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{a^{10}}\right )+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)\) |
Input:
Int[(c + d*x^2)^4*ArcCot[a*x],x]
Output:
c^4*x*ArcCot[a*x] + (4*c^3*d*x^3*ArcCot[a*x])/3 + (6*c^2*d^2*x^5*ArcCot[a* x])/5 + (4*c*d^3*x^7*ArcCot[a*x])/7 + (d^4*x^9*ArcCot[a*x])/9 + (a*((d*(42 0*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/a^8 + (d^2*(378*a ^4*c^2 - 180*a^2*c*d + 35*d^2)*x^4)/(2*a^6) + (5*(36*a^2*c - 7*d)*d^3*x^6) /(3*a^4) + (35*d^4*x^8)/(4*a^2) + ((315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4* c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/a^10))/630
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
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])
Time = 0.54 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00
| method | result | size |
| parts | \(\frac {d^{4} x^{9} \operatorname {arccot}\left (a x \right )}{9}+\frac {4 c \,d^{3} x^{7} \operatorname {arccot}\left (a x \right )}{7}+\frac {6 c^{2} d^{2} x^{5} \operatorname {arccot}\left (a x \right )}{5}+\frac {4 c^{3} d \,x^{3} \operatorname {arccot}\left (a x \right )}{3}+c^{4} x \,\operatorname {arccot}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {35}{4} a^{6} d^{3} x^{8}+60 a^{6} c \,d^{2} x^{6}+189 a^{6} c^{2} d \,x^{4}+420 a^{6} c^{3} x^{2}-\frac {35}{3} a^{4} d^{3} x^{6}-90 a^{4} c \,d^{2} x^{4}-378 a^{4} c^{2} d \,x^{2}+\frac {35}{2} a^{2} d^{3} x^{4}+180 a^{2} c \,d^{2} x^{2}-35 d^{3} x^{2}\right )}{2 a^{8}}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{10}}\right )}{315}\) | \(245\) |
| derivativedivides | \(\frac {\operatorname {arccot}\left (a x \right ) c^{4} a x +\frac {4 a \,\operatorname {arccot}\left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccot}\left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccot}\left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{4} x^{9}}{9}+\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\) | \(254\) |
| default | \(\frac {\operatorname {arccot}\left (a x \right ) c^{4} a x +\frac {4 a \,\operatorname {arccot}\left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccot}\left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccot}\left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{4} x^{9}}{9}+\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\) | \(254\) |
| parallelrisch | \(\frac {840 x^{9} \operatorname {arccot}\left (a x \right ) a^{9} d^{4}+4320 x^{7} \operatorname {arccot}\left (a x \right ) a^{9} c \,d^{3}+105 d^{4} a^{8} x^{8}+9072 x^{5} \operatorname {arccot}\left (a x \right ) a^{9} c^{2} d^{2}+720 c \,a^{8} d^{3} x^{6}+10080 x^{3} \operatorname {arccot}\left (a x \right ) a^{9} c^{3} d -140 d^{4} a^{6} x^{6}+2268 c^{2} a^{8} d^{2} x^{4}+7560 x \,\operatorname {arccot}\left (a x \right ) a^{9} c^{4}-1080 a^{6} c \,d^{3} x^{4}+5040 c^{3} a^{8} d \,x^{2}+3780 \ln \left (a^{2} x^{2}+1\right ) a^{8} c^{4}+210 d^{4} a^{4} x^{4}-4536 c^{2} a^{6} d^{2} x^{2}-5040 \ln \left (a^{2} x^{2}+1\right ) a^{6} c^{3} d +2160 a^{4} c \,d^{3} x^{2}+4536 \ln \left (a^{2} x^{2}+1\right ) a^{4} c^{2} d^{2}-420 d^{4} a^{2} x^{2}-2160 \ln \left (a^{2} x^{2}+1\right ) a^{2} c \,d^{3}+420 \ln \left (a^{2} x^{2}+1\right ) d^{4}}{7560 a^{9}}\) | \(297\) |
| risch | \(-\frac {2 i c \,d^{3} x^{7} \ln \left (-i a x +1\right )}{7}-\frac {3 i c^{2} d^{2} x^{5} \ln \left (-i a x +1\right )}{5}-\frac {2 i c^{3} d \,x^{3} \ln \left (-i a x +1\right )}{3}+\frac {2 c \,d^{3} x^{6}}{21 a}+\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {2 c^{3} d \,x^{2}}{3 a}-\frac {c \,d^{3} x^{4}}{7 a^{3}}-\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}-\frac {i c^{4} x \ln \left (-i a x +1\right )}{2}-\frac {i d^{4} x^{9} \ln \left (-i a x +1\right )}{18}+\frac {2 c \,d^{3} x^{2}}{7 a^{5}}+\frac {2 \pi c \,d^{3} x^{7}}{7}+\frac {3 \pi \,c^{2} d^{2} x^{5}}{5}+\frac {2 \pi \,c^{3} d \,x^{3}}{3}-\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c^{3} d}{3 a^{3}}+\frac {3 \ln \left (-a^{2} x^{2}-1\right ) c^{2} d^{2}}{5 a^{5}}-\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c \,d^{3}}{7 a^{7}}+\frac {\pi \,d^{4} x^{9}}{18}+\frac {\pi \,c^{4} x}{2}+\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{4}}{2 a}+\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{4}}{18 a^{9}}-\frac {d^{4} x^{6}}{54 a^{3}}+\frac {d^{4} x^{4}}{36 a^{5}}-\frac {d^{4} x^{2}}{18 a^{7}}+\frac {i \left (35 d^{4} x^{9}+180 c \,d^{3} x^{7}+378 c^{2} d^{2} x^{5}+420 c^{3} d \,x^{3}+315 c^{4} x \right ) \ln \left (i a x +1\right )}{630}+\frac {d^{4} x^{8}}{72 a}\) | \(413\) |
Input:
int((d*x^2+c)^4*arccot(a*x),x,method=_RETURNVERBOSE)
Output:
1/9*d^4*x^9*arccot(a*x)+4/7*c*d^3*x^7*arccot(a*x)+6/5*c^2*d^2*x^5*arccot(a *x)+4/3*c^3*d*x^3*arccot(a*x)+c^4*x*arccot(a*x)+1/315*a*(1/2*d/a^8*(35/4*a ^6*d^3*x^8+60*a^6*c*d^2*x^6+189*a^6*c^2*d*x^4+420*a^6*c^3*x^2-35/3*a^4*d^3 *x^6-90*a^4*c*d^2*x^4-378*a^4*c^2*d*x^2+35/2*a^2*d^3*x^4+180*a^2*c*d^2*x^2 -35*d^3*x^2)+1/2*(315*a^8*c^4-420*a^6*c^3*d+378*a^4*c^2*d^2-180*a^2*c*d^3+ 35*d^4)/a^10*ln(a^2*x^2+1))
Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.97 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} - 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} - 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d - 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} - 35 \, a^{2} d^{4}\right )} x^{2} + 24 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \operatorname {arccot}\left (a x\right ) + 12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{7560 \, a^{9}} \] Input:
integrate((d*x^2+c)^4*arccot(a*x),x, algorithm="fricas")
Output:
1/7560*(105*a^8*d^4*x^8 + 20*(36*a^8*c*d^3 - 7*a^6*d^4)*x^6 + 6*(378*a^8*c ^2*d^2 - 180*a^6*c*d^3 + 35*a^4*d^4)*x^4 + 12*(420*a^8*c^3*d - 378*a^6*c^2 *d^2 + 180*a^4*c*d^3 - 35*a^2*d^4)*x^2 + 24*(35*a^9*d^4*x^9 + 180*a^9*c*d^ 3*x^7 + 378*a^9*c^2*d^2*x^5 + 420*a^9*c^3*d*x^3 + 315*a^9*c^4*x)*arccot(a* x) + 12*(315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 3 5*d^4)*log(a^2*x^2 + 1))/a^9
Time = 0.62 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.50 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\begin {cases} c^{4} x \operatorname {acot}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {acot}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {acot}{\left (a x \right )}}{9} + \frac {c^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {2 c^{3} d x^{2}}{3 a} + \frac {3 c^{2} d^{2} x^{4}}{10 a} + \frac {2 c d^{3} x^{6}}{21 a} + \frac {d^{4} x^{8}}{72 a} - \frac {2 c^{3} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} - \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} - \frac {c d^{3} x^{4}}{7 a^{3}} - \frac {d^{4} x^{6}}{54 a^{3}} + \frac {3 c^{2} d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{5 a^{5}} + \frac {2 c d^{3} x^{2}}{7 a^{5}} + \frac {d^{4} x^{4}}{36 a^{5}} - \frac {2 c d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{7 a^{7}} - \frac {d^{4} x^{2}}{18 a^{7}} + \frac {d^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{18 a^{9}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{4} x + \frac {4 c^{3} d x^{3}}{3} + \frac {6 c^{2} d^{2} x^{5}}{5} + \frac {4 c d^{3} x^{7}}{7} + \frac {d^{4} x^{9}}{9}\right )}{2} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**2+c)**4*acot(a*x),x)
Output:
Piecewise((c**4*x*acot(a*x) + 4*c**3*d*x**3*acot(a*x)/3 + 6*c**2*d**2*x**5 *acot(a*x)/5 + 4*c*d**3*x**7*acot(a*x)/7 + d**4*x**9*acot(a*x)/9 + c**4*lo g(x**2 + a**(-2))/(2*a) + 2*c**3*d*x**2/(3*a) + 3*c**2*d**2*x**4/(10*a) + 2*c*d**3*x**6/(21*a) + d**4*x**8/(72*a) - 2*c**3*d*log(x**2 + a**(-2))/(3* a**3) - 3*c**2*d**2*x**2/(5*a**3) - c*d**3*x**4/(7*a**3) - d**4*x**6/(54*a **3) + 3*c**2*d**2*log(x**2 + a**(-2))/(5*a**5) + 2*c*d**3*x**2/(7*a**5) + d**4*x**4/(36*a**5) - 2*c*d**3*log(x**2 + a**(-2))/(7*a**7) - d**4*x**2/( 18*a**7) + d**4*log(x**2 + a**(-2))/(18*a**9), Ne(a, 0)), (pi*(c**4*x + 4* c**3*d*x**3/3 + 6*c**2*d**2*x**5/5 + 4*c*d**3*x**7/7 + d**4*x**9/9)/2, Tru e))
Time = 0.02 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.93 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} - 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} - 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d - 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} - 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname {arccot}\left (a x\right ) \] Input:
integrate((d*x^2+c)^4*arccot(a*x),x, algorithm="maxima")
Output:
1/7560*a*((105*a^6*d^4*x^8 + 20*(36*a^6*c*d^3 - 7*a^4*d^4)*x^6 + 6*(378*a^ 6*c^2*d^2 - 180*a^4*c*d^3 + 35*a^2*d^4)*x^4 + 12*(420*a^6*c^3*d - 378*a^4* c^2*d^2 + 180*a^2*c*d^3 - 35*d^4)*x^2)/a^8 + 12*(315*a^8*c^4 - 420*a^6*c^3 *d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*log(a^2*x^2 + 1)/a^10) + 1/ 315*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315*c^ 4*x)*arccot(a*x)
Time = 0.15 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.42 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {1}{7560} \, {\left (\frac {24 \, {\left (35 \, d^{4} + \frac {180 \, c d^{3}}{x^{2}} + \frac {378 \, c^{2} d^{2}}{x^{4}} + \frac {420 \, c^{3} d}{x^{6}} + \frac {315 \, c^{4}}{x^{8}}\right )} x^{9} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (105 \, d^{4} + \frac {720 \, c d^{3}}{x^{2}} + \frac {2268 \, c^{2} d^{2}}{x^{4}} - \frac {140 \, d^{4}}{a^{2} x^{2}} + \frac {5040 \, c^{3} d}{x^{6}} - \frac {1080 \, c d^{3}}{a^{2} x^{4}} + \frac {7875 \, c^{4}}{x^{8}} - \frac {4536 \, c^{2} d^{2}}{a^{2} x^{6}} + \frac {210 \, d^{4}}{a^{4} x^{4}} - \frac {10500 \, c^{3} d}{a^{2} x^{8}} + \frac {2160 \, c d^{3}}{a^{4} x^{6}} + \frac {9450 \, c^{2} d^{2}}{a^{4} x^{8}} - \frac {420 \, d^{4}}{a^{6} x^{6}} - \frac {4500 \, c d^{3}}{a^{6} x^{8}} + \frac {875 \, d^{4}}{a^{8} x^{8}}\right )} x^{8}}{a^{2}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{10}} - \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{10}}\right )} a \] Input:
integrate((d*x^2+c)^4*arccot(a*x),x, algorithm="giac")
Output:
1/7560*(24*(35*d^4 + 180*c*d^3/x^2 + 378*c^2*d^2/x^4 + 420*c^3*d/x^6 + 315 *c^4/x^8)*x^9*arctan(1/(a*x))/a + (105*d^4 + 720*c*d^3/x^2 + 2268*c^2*d^2/ x^4 - 140*d^4/(a^2*x^2) + 5040*c^3*d/x^6 - 1080*c*d^3/(a^2*x^4) + 7875*c^4 /x^8 - 4536*c^2*d^2/(a^2*x^6) + 210*d^4/(a^4*x^4) - 10500*c^3*d/(a^2*x^8) + 2160*c*d^3/(a^4*x^6) + 9450*c^2*d^2/(a^4*x^8) - 420*d^4/(a^6*x^6) - 4500 *c*d^3/(a^6*x^8) + 875*d^4/(a^8*x^8))*x^8/a^2 + 12*(315*a^8*c^4 - 420*a^6* c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*log(1/(a^2*x^2) + 1)/a^1 0 - 12*(315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35 *d^4)*log(1/(a^2*x^2))/a^10)*a
Time = 0.91 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.96 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\mathrm {acot}\left (a\,x\right )\,\left (c^4\,x+\frac {4\,c^3\,d\,x^3}{3}+\frac {6\,c^2\,d^2\,x^5}{5}+\frac {4\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{9}\right )-x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}-\frac {2\,c^3\,d}{3\,a}\right )-x^6\,\left (\frac {d^4}{54\,a^3}-\frac {2\,c\,d^3}{21\,a}\right )+x^4\,\left (\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )+\frac {\ln \left (a^2\,x^2+1\right )\,\left (315\,a^8\,c^4-420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2-180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}+\frac {d^4\,x^8}{72\,a} \] Input:
int(acot(a*x)*(c + d*x^2)^4,x)
Output:
acot(a*x)*(c^4*x + (d^4*x^9)/9 + (4*c^3*d*x^3)/3 + (4*c*d^3*x^7)/7 + (6*c^ 2*d^2*x^5)/5) - x^2*(((d^4/(9*a^3) - (4*c*d^3)/(7*a))/a^2 + (6*c^2*d^2)/(5 *a))/(2*a^2) - (2*c^3*d)/(3*a)) - x^6*(d^4/(54*a^3) - (2*c*d^3)/(21*a)) + x^4*((d^4/(9*a^3) - (4*c*d^3)/(7*a))/(4*a^2) + (3*c^2*d^2)/(10*a)) + (log( a^2*x^2 + 1)*(35*d^4 + 315*a^8*c^4 - 180*a^2*c*d^3 - 420*a^6*c^3*d + 378*a ^4*c^2*d^2))/(630*a^9) + (d^4*x^8)/(72*a)
Time = 0.24 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.21 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {7560 \mathit {acot} \left (a x \right ) a^{9} c^{4} x +10080 \mathit {acot} \left (a x \right ) a^{9} c^{3} d \,x^{3}+9072 \mathit {acot} \left (a x \right ) a^{9} c^{2} d^{2} x^{5}+4320 \mathit {acot} \left (a x \right ) a^{9} c \,d^{3} x^{7}+840 \mathit {acot} \left (a x \right ) a^{9} d^{4} x^{9}+3780 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{8} c^{4}-5040 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{6} c^{3} d +4536 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{4} c^{2} d^{2}-2160 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{2} c \,d^{3}+420 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) d^{4}+5040 a^{8} c^{3} d \,x^{2}+2268 a^{8} c^{2} d^{2} x^{4}+720 a^{8} c \,d^{3} x^{6}+105 a^{8} d^{4} x^{8}-4536 a^{6} c^{2} d^{2} x^{2}-1080 a^{6} c \,d^{3} x^{4}-140 a^{6} d^{4} x^{6}+2160 a^{4} c \,d^{3} x^{2}+210 a^{4} d^{4} x^{4}-420 a^{2} d^{4} x^{2}}{7560 a^{9}} \] Input:
int((d*x^2+c)^4*acot(a*x),x)
Output:
(7560*acot(a*x)*a**9*c**4*x + 10080*acot(a*x)*a**9*c**3*d*x**3 + 9072*acot (a*x)*a**9*c**2*d**2*x**5 + 4320*acot(a*x)*a**9*c*d**3*x**7 + 840*acot(a*x )*a**9*d**4*x**9 + 3780*log(a**2*x**2 + 1)*a**8*c**4 - 5040*log(a**2*x**2 + 1)*a**6*c**3*d + 4536*log(a**2*x**2 + 1)*a**4*c**2*d**2 - 2160*log(a**2* x**2 + 1)*a**2*c*d**3 + 420*log(a**2*x**2 + 1)*d**4 + 5040*a**8*c**3*d*x** 2 + 2268*a**8*c**2*d**2*x**4 + 720*a**8*c*d**3*x**6 + 105*a**8*d**4*x**8 - 4536*a**6*c**2*d**2*x**2 - 1080*a**6*c*d**3*x**4 - 140*a**6*d**4*x**6 + 2 160*a**4*c*d**3*x**2 + 210*a**4*d**4*x**4 - 420*a**2*d**4*x**2)/(7560*a**9 )