3.53 \(\int (c+d x^2)^4 \cot ^{-1}(a x) \, dx\)

Optimal. Leaf size=244 \[ \frac {d^3 x^6 \left (36 a^2 c-7 d\right )}{378 a^3}+\frac {d^2 x^4 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right )}{1260 a^5}+\frac {d x^2 \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right )}{630 a^7}+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{630 a^9}+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac {d^4 x^8}{72 a} \]

[Out]

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*(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)*ln(a^2*x^2+1)/a^9

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Rubi [A]  time = 0.18, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {194, 4913, 1810, 260} \[ \frac {d^2 x^4 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right )}{1260 a^5}+\frac {d x^2 \left (-378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2-35 d^3\right )}{630 a^7}+\frac {\left (378 a^4 c^2 d^2-420 a^6 c^3 d+315 a^8 c^4-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{630 a^9}+\frac {d^3 x^6 \left (36 a^2 c-7 d\right )}{378 a^3}+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+c^4 x \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {d^4 x^8}{72 a}+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4*ArcCot[a*x],x]

[Out]

(d*(420*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 - 180*a^2*c*d + 3
5*d^2)*x^4)/(1260*a^5) + ((36*a^2*c - 7*d)*d^3*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + 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 +
((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])/(630*a^9)

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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])

Rubi steps

\begin {align*} \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, 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)+a \int \frac {c^4 x+\frac {4}{3} c^3 d x^3+\frac {6}{5} c^2 d^2 x^5+\frac {4}{7} c d^3 x^7+\frac {d^4 x^9}{9}}{1+a^2 x^2} \, 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)+a \int \left (\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}{315 a^8}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^3}{315 a^6}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^5}{63 a^4}+\frac {d^4 x^7}{9 a^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 ) x}{315 a^8 \left (1+a^2 x^2\right )}\right ) \, 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 ) \int \frac {x}{1+a^2 x^2} \, dx}{315 a^7}\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 212, normalized size = 0.87 \[ \frac {24 a^9 x \cot ^{-1}(a x) \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )+a^2 d x^2 \left (3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )-4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+30 a^2 d^2 \left (72 c+7 d x^2\right )-420 d^3\right )+12 \left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{7560 a^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^4*ArcCot[a*x],x]

[Out]

(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 + 270*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(16
80*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 +
180*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)

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fricas [A]  time = 0.49, size = 237, normalized size = 0.97 \[ \frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} - 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} - 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d - 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} - 35 \, a^{2} d^{4}\right )} x^{2} + 24 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \operatorname {arccot}\left (a x\right ) + 12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{7560 \, a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arccot(a*x),x, algorithm="fricas")

[Out]

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 + 35*d^4)*log(a^2*x^2 + 1))/a^9

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giac [A]  time = 0.15, size = 347, normalized size = 1.42 \[ \frac {1}{7560} \, {\left (\frac {24 \, {\left (35 \, d^{4} + \frac {180 \, c d^{3}}{x^{2}} + \frac {378 \, c^{2} d^{2}}{x^{4}} + \frac {420 \, c^{3} d}{x^{6}} + \frac {315 \, c^{4}}{x^{8}}\right )} x^{9} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (105 \, d^{4} + \frac {720 \, c d^{3}}{x^{2}} + \frac {2268 \, c^{2} d^{2}}{x^{4}} - \frac {140 \, d^{4}}{a^{2} x^{2}} + \frac {5040 \, c^{3} d}{x^{6}} - \frac {1080 \, c d^{3}}{a^{2} x^{4}} + \frac {7875 \, c^{4}}{x^{8}} - \frac {4536 \, c^{2} d^{2}}{a^{2} x^{6}} + \frac {210 \, d^{4}}{a^{4} x^{4}} - \frac {10500 \, c^{3} d}{a^{2} x^{8}} + \frac {2160 \, c d^{3}}{a^{4} x^{6}} + \frac {9450 \, c^{2} d^{2}}{a^{4} x^{8}} - \frac {420 \, d^{4}}{a^{6} x^{6}} - \frac {4500 \, c d^{3}}{a^{6} x^{8}} + \frac {875 \, d^{4}}{a^{8} x^{8}}\right )} x^{8}}{a^{2}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{10}} - \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{10}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arccot(a*x),x, algorithm="giac")

[Out]

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 + (1
05*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^10 - 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

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maple [A]  time = 0.04, size = 279, normalized size = 1.14 \[ \frac {d^{4} x^{9} \mathrm {arccot}\left (a x \right )}{9}+\frac {4 c \,d^{3} x^{7} \mathrm {arccot}\left (a x \right )}{7}+\frac {6 c^{2} d^{2} x^{5} \mathrm {arccot}\left (a x \right )}{5}+\frac {4 c^{3} d \,x^{3} \mathrm {arccot}\left (a x \right )}{3}+c^{4} x \,\mathrm {arccot}\left (a x \right )+\frac {2 c^{3} d \,x^{2}}{3 a}+\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {2 c \,d^{3} x^{6}}{21 a}-\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}+\frac {d^{4} x^{8}}{72 a}-\frac {x^{4} c \,d^{3}}{7 a^{3}}-\frac {d^{4} x^{6}}{54 a^{3}}+\frac {2 x^{2} c \,d^{3}}{7 a^{5}}+\frac {d^{4} x^{4}}{36 a^{5}}-\frac {x^{2} d^{4}}{18 a^{7}}+\frac {\ln \left (a^{2} x^{2}+1\right ) c^{4}}{2 a}-\frac {2 \ln \left (a^{2} x^{2}+1\right ) c^{3} d}{3 a^{3}}+\frac {3 \ln \left (a^{2} x^{2}+1\right ) c^{2} d^{2}}{5 a^{5}}-\frac {2 \ln \left (a^{2} x^{2}+1\right ) c \,d^{3}}{7 a^{7}}+\frac {\ln \left (a^{2} x^{2}+1\right ) d^{4}}{18 a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^4*arccot(a*x),x)

[Out]

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)+2/3/a*c^3*d*x^2+3/10/a*c^2*d^2*x^4+2/21/a*c*d^3*x^6-3/5/a^3*c^2*d^2*x^2+1/72*d^4*x^8/a-1/7/a^3*x^4
*c*d^3-1/54/a^3*d^4*x^6+2/7/a^5*x^2*c*d^3+1/36/a^5*d^4*x^4-1/18/a^7*x^2*d^4+1/2/a*ln(a^2*x^2+1)*c^4-2/3/a^3*ln
(a^2*x^2+1)*c^3*d+3/5/a^5*ln(a^2*x^2+1)*c^2*d^2-2/7/a^7*ln(a^2*x^2+1)*c*d^3+1/18/a^9*ln(a^2*x^2+1)*d^4

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maxima [A]  time = 0.32, size = 226, normalized size = 0.93 \[ \frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} - 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} - 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d - 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} - 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname {arccot}\left (a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arccot(a*x),x, algorithm="maxima")

[Out]

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)

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mupad [B]  time = 0.22, size = 234, normalized size = 0.96 \[ \mathrm {acot}\left (a\,x\right )\,\left (c^4\,x+\frac {4\,c^3\,d\,x^3}{3}+\frac {6\,c^2\,d^2\,x^5}{5}+\frac {4\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{9}\right )-x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}-\frac {2\,c^3\,d}{3\,a}\right )-x^6\,\left (\frac {d^4}{54\,a^3}-\frac {2\,c\,d^3}{21\,a}\right )+x^4\,\left (\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )+\frac {\ln \left (a^2\,x^2+1\right )\,\left (315\,a^8\,c^4-420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2-180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}+\frac {d^4\,x^8}{72\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)*(c + d*x^2)^4,x)

[Out]

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)

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sympy [A]  time = 4.36, size = 367, normalized size = 1.50 \[ \begin {cases} c^{4} x \operatorname {acot}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {acot}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {acot}{\left (a x \right )}}{9} + \frac {c^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {2 c^{3} d x^{2}}{3 a} + \frac {3 c^{2} d^{2} x^{4}}{10 a} + \frac {2 c d^{3} x^{6}}{21 a} + \frac {d^{4} x^{8}}{72 a} - \frac {2 c^{3} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} - \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} - \frac {c d^{3} x^{4}}{7 a^{3}} - \frac {d^{4} x^{6}}{54 a^{3}} + \frac {3 c^{2} d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{5 a^{5}} + \frac {2 c d^{3} x^{2}}{7 a^{5}} + \frac {d^{4} x^{4}}{36 a^{5}} - \frac {2 c d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{7 a^{7}} - \frac {d^{4} x^{2}}{18 a^{7}} + \frac {d^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{18 a^{9}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{4} x + \frac {4 c^{3} d x^{3}}{3} + \frac {6 c^{2} d^{2} x^{5}}{5} + \frac {4 c d^{3} x^{7}}{7} + \frac {d^{4} x^{9}}{9}\right )}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**4*acot(a*x),x)

[Out]

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*log(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, True))

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