Integrand size = 14, antiderivative size = 109 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {\left (10 a^2 c-3 d\right ) d x^2}{30 a^3}+\frac {d^2 x^4}{20 a}+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac {\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (1+a^2 x^2\right )}{30 a^5} \] Output:
1/30*(10*a^2*c-3*d)*d*x^2/a^3+1/20*d^2*x^4/a+c^2*x*arccot(a*x)+2/3*c*d*x^3 *arccot(a*x)+1/5*d^2*x^5*arccot(a*x)+1/30*(15*a^4*c^2-10*a^2*c*d+3*d^2)*ln (a^2*x^2+1)/a^5
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (-6 d+a^2 \left (20 c+3 d x^2\right )\right )+4 a^5 x \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \cot ^{-1}(a x)+\left (30 a^4 c^2-20 a^2 c d+6 d^2\right ) \log \left (1+a^2 x^2\right )}{60 a^5} \] Input:
Integrate[(c + d*x^2)^2*ArcCot[a*x],x]
Output:
(a^2*d*x^2*(-6*d + a^2*(20*c + 3*d*x^2)) + 4*a^5*x*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*ArcCot[a*x] + (30*a^4*c^2 - 20*a^2*c*d + 6*d^2)*Log[1 + a^2*x^2 ])/(60*a^5)
Time = 0.36 (sec) , antiderivative size = 109, 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, 1576, 1140, 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.
| \(\displaystyle \int \cot ^{-1}(a x) \left (c+d x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 5448 |
| \(\displaystyle a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{15 \left (a^2 x^2+1\right )}dx+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{a^2 x^2+1}dx+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{30} a \int \frac {3 d^2 x^4+10 c d x^2+15 c^2}{a^2 x^2+1}dx^2+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {1}{30} a \int \left (\frac {3 d^2 x^2}{a^2}+\frac {\left (10 a^2 c-3 d\right ) d}{a^4}+\frac {15 c^2 a^4-10 c d a^2+3 d^2}{a^4 \left (a^2 x^2+1\right )}\right )dx^2+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{30} a \left (\frac {3 d^2 x^4}{2 a^2}+\frac {d x^2 \left (10 a^2 c-3 d\right )}{a^4}+\frac {\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (a^2 x^2+1\right )}{a^6}\right )+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)\) |
Input:
Int[(c + d*x^2)^2*ArcCot[a*x],x]
Output:
c^2*x*ArcCot[a*x] + (2*c*d*x^3*ArcCot[a*x])/3 + (d^2*x^5*ArcCot[a*x])/5 + (a*(((10*a^2*c - 3*d)*d*x^2)/a^4 + (3*d^2*x^4)/(2*a^2) + ((15*a^4*c^2 - 10 *a^2*c*d + 3*d^2)*Log[1 + a^2*x^2])/a^6))/30
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
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 = 0.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.96
| method | result | size |
| parts | \(\frac {d^{2} x^{5} \operatorname {arccot}\left (a x \right )}{5}+\frac {2 c d \,x^{3} \operatorname {arccot}\left (a x \right )}{3}+c^{2} x \,\operatorname {arccot}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {3}{2} a^{2} d \,x^{4}+10 a^{2} c \,x^{2}-3 d \,x^{2}\right )}{2 a^{4}}+\frac {\left (15 a^{4} c^{2}-10 a^{2} c d +3 d^{2}\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{6}}\right )}{15}\) | \(105\) |
| derivativedivides | \(\frac {\operatorname {arccot}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccot}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{2} x^{5}}{5}+\frac {5 c \,a^{4} d \,x^{2}+\frac {3 d^{2} a^{4} x^{4}}{4}-\frac {3 a^{2} d^{2} x^{2}}{2}+\frac {\left (15 a^{4} c^{2}-10 a^{2} c d +3 d^{2}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{15 a^{4}}}{a}\) | \(112\) |
| default | \(\frac {\operatorname {arccot}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccot}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{2} x^{5}}{5}+\frac {5 c \,a^{4} d \,x^{2}+\frac {3 d^{2} a^{4} x^{4}}{4}-\frac {3 a^{2} d^{2} x^{2}}{2}+\frac {\left (15 a^{4} c^{2}-10 a^{2} c d +3 d^{2}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{15 a^{4}}}{a}\) | \(112\) |
| parallelrisch | \(\frac {12 x^{5} \operatorname {arccot}\left (a x \right ) a^{5} d^{2}+40 x^{3} \operatorname {arccot}\left (a x \right ) a^{5} c d +3 d^{2} a^{4} x^{4}+60 c^{2} \operatorname {arccot}\left (a x \right ) x \,a^{5}+20 c \,a^{4} d \,x^{2}+30 \ln \left (a^{2} x^{2}+1\right ) a^{4} c^{2}-6 a^{2} d^{2} x^{2}-20 \ln \left (a^{2} x^{2}+1\right ) a^{2} c d -20 a^{2} c d +6 \ln \left (a^{2} x^{2}+1\right ) d^{2}+6 d^{2}}{60 a^{5}}\) | \(143\) |
| risch | \(\frac {i \left (3 d^{2} x^{5}+10 c d \,x^{3}+15 c^{2} x \right ) \ln \left (i a x +1\right )}{30}-\frac {i d^{2} x^{5} \ln \left (-i a x +1\right )}{10}+\frac {\pi \,d^{2} x^{5}}{10}-\frac {i c d \,x^{3} \ln \left (-i a x +1\right )}{3}+\frac {\pi c d \,x^{3}}{3}-\frac {i c^{2} x \ln \left (-i a x +1\right )}{2}+\frac {d^{2} x^{4}}{20 a}+\frac {\pi \,c^{2} x}{2}+\frac {c d \,x^{2}}{3 a}+\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{2}}{2 a}-\frac {d^{2} x^{2}}{10 a^{3}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) c d}{3 a^{3}}+\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{2}}{10 a^{5}}\) | \(195\) |
Input:
int((d*x^2+c)^2*arccot(a*x),x,method=_RETURNVERBOSE)
Output:
1/5*d^2*x^5*arccot(a*x)+2/3*c*d*x^3*arccot(a*x)+c^2*x*arccot(a*x)+1/15*a*( 1/2*d/a^4*(3/2*a^2*d*x^4+10*a^2*c*x^2-3*d*x^2)+1/2*(15*a^4*c^2-10*a^2*c*d+ 3*d^2)/a^6*ln(a^2*x^2+1))
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.99 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {3 \, a^{4} d^{2} x^{4} + 2 \, {\left (10 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} + 4 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \operatorname {arccot}\left (a x\right ) + 2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{60 \, a^{5}} \] Input:
integrate((d*x^2+c)^2*arccot(a*x),x, algorithm="fricas")
Output:
1/60*(3*a^4*d^2*x^4 + 2*(10*a^4*c*d - 3*a^2*d^2)*x^2 + 4*(3*a^5*d^2*x^5 + 10*a^5*c*d*x^3 + 15*a^5*c^2*x)*arccot(a*x) + 2*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(a^2*x^2 + 1))/a^5
Time = 0.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.39 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\begin {cases} c^{2} x \operatorname {acot}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {c d x^{2}}{3 a} + \frac {d^{2} x^{4}}{20 a} - \frac {c d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} - \frac {d^{2} x^{2}}{10 a^{3}} + \frac {d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{10 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**2+c)**2*acot(a*x),x)
Output:
Piecewise((c**2*x*acot(a*x) + 2*c*d*x**3*acot(a*x)/3 + d**2*x**5*acot(a*x) /5 + c**2*log(x**2 + a**(-2))/(2*a) + c*d*x**2/(3*a) + d**2*x**4/(20*a) - c*d*log(x**2 + a**(-2))/(3*a**3) - d**2*x**2/(10*a**3) + d**2*log(x**2 + a **(-2))/(10*a**5), Ne(a, 0)), (pi*(c**2*x + 2*c*d*x**3/3 + d**2*x**5/5)/2, True))
Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {1}{60} \, a {\left (\frac {3 \, a^{2} d^{2} x^{4} + 2 \, {\left (10 \, a^{2} c d - 3 \, d^{2}\right )} x^{2}}{a^{4}} + \frac {2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arccot}\left (a x\right ) \] Input:
integrate((d*x^2+c)^2*arccot(a*x),x, algorithm="maxima")
Output:
1/60*a*((3*a^2*d^2*x^4 + 2*(10*a^2*c*d - 3*d^2)*x^2)/a^4 + 2*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(a^2*x^2 + 1)/a^6) + 1/15*(3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*arccot(a*x)
Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.57 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {1}{60} \, {\left (\frac {4 \, {\left (3 \, d^{2} + \frac {10 \, c d}{x^{2}} + \frac {15 \, c^{2}}{x^{4}}\right )} x^{5} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (3 \, d^{2} + \frac {20 \, c d}{x^{2}} + \frac {45 \, c^{2}}{x^{4}} - \frac {6 \, d^{2}}{a^{2} x^{2}} - \frac {30 \, c d}{a^{2} x^{4}} + \frac {9 \, d^{2}}{a^{4} x^{4}}\right )} x^{4}}{a^{2}} + \frac {2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{6}} - \frac {2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{6}}\right )} a \] Input:
integrate((d*x^2+c)^2*arccot(a*x),x, algorithm="giac")
Output:
1/60*(4*(3*d^2 + 10*c*d/x^2 + 15*c^2/x^4)*x^5*arctan(1/(a*x))/a + (3*d^2 + 20*c*d/x^2 + 45*c^2/x^4 - 6*d^2/(a^2*x^2) - 30*c*d/(a^2*x^4) + 9*d^2/(a^4 *x^4))*x^4/a^2 + 2*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(1/(a^2*x^2) + 1)/ a^6 - 2*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(1/(a^2*x^2))/a^6)*a
Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {a^4\,\left (\frac {c^2\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {d^2\,x^4}{20}+\frac {c\,d\,x^2}{3}\right )-a^2\,\left (\frac {d^2\,x^2}{10}+\frac {c\,d\,\ln \left (a^2\,x^2+1\right )}{3}\right )+\frac {d^2\,\ln \left (a^2\,x^2+1\right )}{10}}{a^5}+c^2\,x\,\mathrm {acot}\left (a\,x\right )+\frac {d^2\,x^5\,\mathrm {acot}\left (a\,x\right )}{5}+\frac {2\,c\,d\,x^3\,\mathrm {acot}\left (a\,x\right )}{3} \] Input:
int(acot(a*x)*(c + d*x^2)^2,x)
Output:
(a^4*((c^2*log(a^2*x^2 + 1))/2 + (d^2*x^4)/20 + (c*d*x^2)/3) - a^2*((d^2*x ^2)/10 + (c*d*log(a^2*x^2 + 1))/3) + (d^2*log(a^2*x^2 + 1))/10)/a^5 + c^2* x*acot(a*x) + (d^2*x^5*acot(a*x))/5 + (2*c*d*x^3*acot(a*x))/3
Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.19 \[ \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx=\frac {60 \mathit {acot} \left (a x \right ) a^{5} c^{2} x +40 \mathit {acot} \left (a x \right ) a^{5} c d \,x^{3}+12 \mathit {acot} \left (a x \right ) a^{5} d^{2} x^{5}+30 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{4} c^{2}-20 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{2} c d +6 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) d^{2}+20 a^{4} c d \,x^{2}+3 a^{4} d^{2} x^{4}-6 a^{2} d^{2} x^{2}}{60 a^{5}} \] Input:
int((d*x^2+c)^2*acot(a*x),x)
Output:
(60*acot(a*x)*a**5*c**2*x + 40*acot(a*x)*a**5*c*d*x**3 + 12*acot(a*x)*a**5 *d**2*x**5 + 30*log(a**2*x**2 + 1)*a**4*c**2 - 20*log(a**2*x**2 + 1)*a**2* c*d + 6*log(a**2*x**2 + 1)*d**2 + 20*a**4*c*d*x**2 + 3*a**4*d**2*x**4 - 6* a**2*d**2*x**2)/(60*a**5)