Integrand size = 14, antiderivative size = 168 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac {\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac {d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{70 a^7} \] Output:
1/70*d*(35*a^4*c^2-21*a^2*c*d+5*d^2)*x^2/a^5+1/140*(21*a^2*c-5*d)*d^2*x^4/ a^3+1/42*d^3*x^6/a+c^3*x*arccot(a*x)+c^2*d*x^3*arccot(a*x)+3/5*c*d^2*x^5*a rccot(a*x)+1/7*d^3*x^7*arccot(a*x)+1/70*(35*a^6*c^3-35*a^4*c^2*d+21*a^2*c* d^2-5*d^3)*ln(a^2*x^2+1)/a^7
Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.89 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (30 d^2-3 a^2 d \left (42 c+5 d x^2\right )+a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )\right )+12 a^7 x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \cot ^{-1}(a x)+6 \left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{420 a^7} \] Input:
Integrate[(c + d*x^2)^3*ArcCot[a*x],x]
Output:
(a^2*d*x^2*(30*d^2 - 3*a^2*d*(42*c + 5*d*x^2) + a^4*(210*c^2 + 63*c*d*x^2 + 10*d^2*x^4)) + 12*a^7*x*(35*c^3 + 35*c^2*d*x^2 + 21*c*d^2*x^4 + 5*d^3*x^ 6)*ArcCot[a*x] + 6*(35*a^6*c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[ 1 + a^2*x^2])/(420*a^7)
Time = 0.53 (sec) , antiderivative size = 168, 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.
| \(\displaystyle \int \cot ^{-1}(a x) \left (c+d x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 5448 |
| \(\displaystyle a \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{35 \left (a^2 x^2+1\right )}dx+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} a \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{a^2 x^2+1}dx+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{70} a \int \frac {5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3}{a^2 x^2+1}dx^2+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {1}{70} a \int \left (\frac {5 d^3 x^4}{a^2}+\frac {\left (21 a^2 c-5 d\right ) d^2 x^2}{a^4}+\frac {d \left (35 c^2 a^4-21 c d a^2+5 d^2\right )}{a^6}+\frac {35 c^3 a^6-35 c^2 d a^4+21 c d^2 a^2-5 d^3}{a^6 \left (a^2 x^2+1\right )}\right )dx^2+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{70} a \left (\frac {5 d^3 x^6}{3 a^2}+\frac {d^2 x^4 \left (21 a^2 c-5 d\right )}{2 a^4}+\frac {d x^2 \left (35 a^4 c^2-21 a^2 c d+5 d^2\right )}{a^6}+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )}{a^8}\right )+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)\) |
Input:
Int[(c + d*x^2)^3*ArcCot[a*x],x]
Output:
c^3*x*ArcCot[a*x] + c^2*d*x^3*ArcCot[a*x] + (3*c*d^2*x^5*ArcCot[a*x])/5 + (d^3*x^7*ArcCot[a*x])/7 + (a*((d*(35*a^4*c^2 - 21*a^2*c*d + 5*d^2)*x^2)/a^ 6 + ((21*a^2*c - 5*d)*d^2*x^4)/(2*a^4) + (5*d^3*x^6)/(3*a^2) + ((35*a^6*c^ 3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[1 + a^2*x^2])/a^8))/70
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 = 0.42 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99
| method | result | size |
| parts | \(\frac {d^{3} x^{7} \operatorname {arccot}\left (a x \right )}{7}+\frac {3 c \,d^{2} x^{5} \operatorname {arccot}\left (a x \right )}{5}+c^{2} d \,x^{3} \operatorname {arccot}\left (a x \right )+c^{3} x \,\operatorname {arccot}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {5}{3} a^{4} d^{2} x^{6}+\frac {21}{2} a^{4} c d \,x^{4}+35 a^{4} c^{2} x^{2}-\frac {5}{2} a^{2} d^{2} x^{4}-21 a^{2} c d \,x^{2}+5 d^{2} x^{2}\right )}{2 a^{6}}+\frac {\left (35 a^{6} c^{3}-35 a^{4} c^{2} d +21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{8}}\right )}{35}\) | \(167\) |
| derivativedivides | \(\frac {\operatorname {arccot}\left (a x \right ) c^{3} a x +a \,\operatorname {arccot}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccot}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{3} x^{7}}{7}+\frac {\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}-\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}-\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {\left (35 a^{6} c^{3}-35 a^{4} c^{2} d +21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{35 a^{6}}}{a}\) | \(175\) |
| default | \(\frac {\operatorname {arccot}\left (a x \right ) c^{3} a x +a \,\operatorname {arccot}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccot}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{3} x^{7}}{7}+\frac {\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}-\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}-\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {\left (35 a^{6} c^{3}-35 a^{4} c^{2} d +21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{35 a^{6}}}{a}\) | \(175\) |
| parallelrisch | \(\frac {60 x^{7} \operatorname {arccot}\left (a x \right ) a^{7} d^{3}+252 x^{5} \operatorname {arccot}\left (a x \right ) a^{7} c \,d^{2}+10 d^{3} a^{6} x^{6}+420 x^{3} \operatorname {arccot}\left (a x \right ) a^{7} c^{2} d +63 c \,a^{6} d^{2} x^{4}+420 x \,\operatorname {arccot}\left (a x \right ) a^{7} c^{3}-15 d^{3} a^{4} x^{4}+210 c^{2} a^{6} d \,x^{2}+210 \ln \left (a^{2} x^{2}+1\right ) a^{6} c^{3}-126 c \,a^{4} d^{2} x^{2}-210 \ln \left (a^{2} x^{2}+1\right ) a^{4} c^{2} d +30 d^{3} a^{2} x^{2}+126 \ln \left (a^{2} x^{2}+1\right ) a^{2} c \,d^{2}-30 \ln \left (a^{2} x^{2}+1\right ) d^{3}}{420 a^{7}}\) | \(207\) |
| risch | \(-\frac {i d^{3} x^{7} \ln \left (-i a x +1\right )}{14}+\frac {i \left (5 d^{3} x^{7}+21 d^{2} c \,x^{5}+35 c^{2} d \,x^{3}+35 c^{3} x \right ) \ln \left (i a x +1\right )}{70}+\frac {\pi \,d^{3} x^{7}}{14}-\frac {i c^{3} x \ln \left (-i a x +1\right )}{2}+\frac {3 \pi c \,d^{2} x^{5}}{10}-\frac {i c^{2} d \,x^{3} \ln \left (-i a x +1\right )}{2}+\frac {d^{3} x^{6}}{42 a}+\frac {\pi \,c^{2} d \,x^{3}}{2}-\frac {3 i c \,d^{2} x^{5} \ln \left (-i a x +1\right )}{10}+\frac {3 c \,d^{2} x^{4}}{20 a}+\frac {\pi \,c^{3} x}{2}+\frac {c^{2} d \,x^{2}}{2 a}-\frac {d^{3} x^{4}}{28 a^{3}}+\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{3}}{2 a}-\frac {3 c \,d^{2} x^{2}}{10 a^{3}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{2} d}{2 a^{3}}+\frac {d^{3} x^{2}}{14 a^{5}}+\frac {3 \ln \left (-a^{2} x^{2}-1\right ) c \,d^{2}}{10 a^{5}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{3}}{14 a^{7}}\) | \(297\) |
Input:
int((d*x^2+c)^3*arccot(a*x),x,method=_RETURNVERBOSE)
Output:
1/7*d^3*x^7*arccot(a*x)+3/5*c*d^2*x^5*arccot(a*x)+c^2*d*x^3*arccot(a*x)+c^ 3*x*arccot(a*x)+1/35*a*(1/2*d/a^6*(5/3*a^4*d^2*x^6+21/2*a^4*c*d*x^4+35*a^4 *c^2*x^2-5/2*a^2*d^2*x^4-21*a^2*c*d*x^2+5*d^2*x^2)+1/2*(35*a^6*c^3-35*a^4* c^2*d+21*a^2*c*d^2-5*d^3)/a^8*ln(a^2*x^2+1))
Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {10 \, a^{6} d^{3} x^{6} + 3 \, {\left (21 \, a^{6} c d^{2} - 5 \, a^{4} d^{3}\right )} x^{4} + 6 \, {\left (35 \, a^{6} c^{2} d - 21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2} + 12 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \operatorname {arccot}\left (a x\right ) + 6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{420 \, a^{7}} \] Input:
integrate((d*x^2+c)^3*arccot(a*x),x, algorithm="fricas")
Output:
1/420*(10*a^6*d^3*x^6 + 3*(21*a^6*c*d^2 - 5*a^4*d^3)*x^4 + 6*(35*a^6*c^2*d - 21*a^4*c*d^2 + 5*a^2*d^3)*x^2 + 12*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35*a^7*c^2*d*x^3 + 35*a^7*c^3*x)*arccot(a*x) + 6*(35*a^6*c^3 - 35*a^4*c^2* d + 21*a^2*c*d^2 - 5*d^3)*log(a^2*x^2 + 1))/a^7
Time = 0.45 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.45 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\begin {cases} c^{3} x \operatorname {acot}{\left (a x \right )} + c^{2} d x^{3} \operatorname {acot}{\left (a x \right )} + \frac {3 c d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {d^{3} x^{7} \operatorname {acot}{\left (a x \right )}}{7} + \frac {c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {c^{2} d x^{2}}{2 a} + \frac {3 c d^{2} x^{4}}{20 a} + \frac {d^{3} x^{6}}{42 a} - \frac {c^{2} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a^{3}} - \frac {3 c d^{2} x^{2}}{10 a^{3}} - \frac {d^{3} x^{4}}{28 a^{3}} + \frac {3 c d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{10 a^{5}} + \frac {d^{3} x^{2}}{14 a^{5}} - \frac {d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{14 a^{7}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}\right )}{2} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**2+c)**3*acot(a*x),x)
Output:
Piecewise((c**3*x*acot(a*x) + c**2*d*x**3*acot(a*x) + 3*c*d**2*x**5*acot(a *x)/5 + d**3*x**7*acot(a*x)/7 + c**3*log(x**2 + a**(-2))/(2*a) + c**2*d*x* *2/(2*a) + 3*c*d**2*x**4/(20*a) + d**3*x**6/(42*a) - c**2*d*log(x**2 + a** (-2))/(2*a**3) - 3*c*d**2*x**2/(10*a**3) - d**3*x**4/(28*a**3) + 3*c*d**2* log(x**2 + a**(-2))/(10*a**5) + d**3*x**2/(14*a**5) - d**3*log(x**2 + a**( -2))/(14*a**7), Ne(a, 0)), (pi*(c**3*x + c**2*d*x**3 + 3*c*d**2*x**5/5 + d **3*x**7/7)/2, True))
Time = 0.02 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {1}{420} \, a {\left (\frac {10 \, a^{4} d^{3} x^{6} + 3 \, {\left (21 \, a^{4} c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 6 \, {\left (35 \, a^{4} c^{2} d - 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} x^{2}}{a^{6}} + \frac {6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{8}}\right )} + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname {arccot}\left (a x\right ) \] Input:
integrate((d*x^2+c)^3*arccot(a*x),x, algorithm="maxima")
Output:
1/420*a*((10*a^4*d^3*x^6 + 3*(21*a^4*c*d^2 - 5*a^2*d^3)*x^4 + 6*(35*a^4*c^ 2*d - 21*a^2*c*d^2 + 5*d^3)*x^2)/a^6 + 6*(35*a^6*c^3 - 35*a^4*c^2*d + 21*a ^2*c*d^2 - 5*d^3)*log(a^2*x^2 + 1)/a^8) + 1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*arccot(a*x)
Time = 0.13 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.50 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {1}{420} \, {\left (\frac {12 \, {\left (5 \, d^{3} + \frac {21 \, c d^{2}}{x^{2}} + \frac {35 \, c^{2} d}{x^{4}} + \frac {35 \, c^{3}}{x^{6}}\right )} x^{7} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (10 \, d^{3} + \frac {63 \, c d^{2}}{x^{2}} + \frac {210 \, c^{2} d}{x^{4}} - \frac {15 \, d^{3}}{a^{2} x^{2}} + \frac {385 \, c^{3}}{x^{6}} - \frac {126 \, c d^{2}}{a^{2} x^{4}} - \frac {385 \, c^{2} d}{a^{2} x^{6}} + \frac {30 \, d^{3}}{a^{4} x^{4}} + \frac {231 \, c d^{2}}{a^{4} x^{6}} - \frac {55 \, d^{3}}{a^{6} x^{6}}\right )} x^{6}}{a^{2}} + \frac {6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{8}} - \frac {6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{8}}\right )} a \] Input:
integrate((d*x^2+c)^3*arccot(a*x),x, algorithm="giac")
Output:
1/420*(12*(5*d^3 + 21*c*d^2/x^2 + 35*c^2*d/x^4 + 35*c^3/x^6)*x^7*arctan(1/ (a*x))/a + (10*d^3 + 63*c*d^2/x^2 + 210*c^2*d/x^4 - 15*d^3/(a^2*x^2) + 385 *c^3/x^6 - 126*c*d^2/(a^2*x^4) - 385*c^2*d/(a^2*x^6) + 30*d^3/(a^4*x^4) + 231*c*d^2/(a^4*x^6) - 55*d^3/(a^6*x^6))*x^6/a^2 + 6*(35*a^6*c^3 - 35*a^4*c ^2*d + 21*a^2*c*d^2 - 5*d^3)*log(1/(a^2*x^2) + 1)/a^8 - 6*(35*a^6*c^3 - 35 *a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*log(1/(a^2*x^2))/a^8)*a
Time = 1.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.13 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=c^3\,x\,\mathrm {acot}\left (a\,x\right )+\frac {d^3\,x^7\,\mathrm {acot}\left (a\,x\right )}{7}+\frac {c^3\,\ln \left (a^2\,x^2+1\right )}{2\,a}-\frac {d^3\,\ln \left (a^2\,x^2+1\right )}{14\,a^7}+\frac {d^3\,x^6}{42\,a}-\frac {d^3\,x^4}{28\,a^3}+\frac {d^3\,x^2}{14\,a^5}-\frac {c^2\,d\,\ln \left (a^2\,x^2+1\right )}{2\,a^3}+\frac {3\,c\,d^2\,\ln \left (a^2\,x^2+1\right )}{10\,a^5}+\frac {c^2\,d\,x^2}{2\,a}+\frac {3\,c\,d^2\,x^4}{20\,a}-\frac {3\,c\,d^2\,x^2}{10\,a^3}+c^2\,d\,x^3\,\mathrm {acot}\left (a\,x\right )+\frac {3\,c\,d^2\,x^5\,\mathrm {acot}\left (a\,x\right )}{5} \] Input:
int(acot(a*x)*(c + d*x^2)^3,x)
Output:
c^3*x*acot(a*x) + (d^3*x^7*acot(a*x))/7 + (c^3*log(a^2*x^2 + 1))/(2*a) - ( d^3*log(a^2*x^2 + 1))/(14*a^7) + (d^3*x^6)/(42*a) - (d^3*x^4)/(28*a^3) + ( d^3*x^2)/(14*a^5) - (c^2*d*log(a^2*x^2 + 1))/(2*a^3) + (3*c*d^2*log(a^2*x^ 2 + 1))/(10*a^5) + (c^2*d*x^2)/(2*a) + (3*c*d^2*x^4)/(20*a) - (3*c*d^2*x^2 )/(10*a^3) + c^2*d*x^3*acot(a*x) + (3*c*d^2*x^5*acot(a*x))/5
Time = 0.24 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.23 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {420 \mathit {acot} \left (a x \right ) a^{7} c^{3} x +420 \mathit {acot} \left (a x \right ) a^{7} c^{2} d \,x^{3}+252 \mathit {acot} \left (a x \right ) a^{7} c \,d^{2} x^{5}+60 \mathit {acot} \left (a x \right ) a^{7} d^{3} x^{7}+210 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{6} c^{3}-210 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{4} c^{2} d +126 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{2} c \,d^{2}-30 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) d^{3}+210 a^{6} c^{2} d \,x^{2}+63 a^{6} c \,d^{2} x^{4}+10 a^{6} d^{3} x^{6}-126 a^{4} c \,d^{2} x^{2}-15 a^{4} d^{3} x^{4}+30 a^{2} d^{3} x^{2}}{420 a^{7}} \] Input:
int((d*x^2+c)^3*acot(a*x),x)
Output:
(420*acot(a*x)*a**7*c**3*x + 420*acot(a*x)*a**7*c**2*d*x**3 + 252*acot(a*x )*a**7*c*d**2*x**5 + 60*acot(a*x)*a**7*d**3*x**7 + 210*log(a**2*x**2 + 1)* a**6*c**3 - 210*log(a**2*x**2 + 1)*a**4*c**2*d + 126*log(a**2*x**2 + 1)*a* *2*c*d**2 - 30*log(a**2*x**2 + 1)*d**3 + 210*a**6*c**2*d*x**2 + 63*a**6*c* d**2*x**4 + 10*a**6*d**3*x**6 - 126*a**4*c*d**2*x**2 - 15*a**4*d**3*x**4 + 30*a**2*d**3*x**2)/(420*a**7)