Integrand size = 12, antiderivative size = 143 \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {3 i b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \]
I*(a+b*arccot(d*x+c))^3/d+(d*x+c)*(a+b*arccot(d*x+c))^3/d-3*b*(a+b*arccot( d*x+c))^2*ln(2/(1+I*(d*x+c)))/d+3*I*b^2*(a+b*arccot(d*x+c))*polylog(2,1-2/ (1+I*(d*x+c)))/d-3/2*b^3*polylog(3,1-2/(1+I*(d*x+c)))/d
Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.59 \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \cot ^{-1}(c+d x)+3 a^2 b \log \left (1+(c+d x)^2\right )+6 a b^2 \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )+2 b^3 \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )-3 i \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )\right )}{2 d} \]
(2*a^3*(c + d*x) + 6*a^2*b*(c + d*x)*ArcCot[c + d*x] + 3*a^2*b*Log[1 + (c + d*x)^2] + 6*a*b^2*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Lo g[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x]) ]) + 2*b^3*((I/8)*Pi^3 - I*ArcCot[c + d*x]^3 + (c + d*x)*ArcCot[c + d*x]^3 - 3*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])] - (3*I)*ArcCot[ c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2*I)* ArcCot[c + d*x])])/2))/(2*d)
Time = 0.67 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5563, 5346, 5456, 5380, 5530, 7164}
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.
| \(\displaystyle \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 5563 |
| \(\displaystyle \frac {\int \left (a+b \cot ^{-1}(c+d x)\right )^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5346 |
| \(\displaystyle \frac {3 b \int \frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{(c+d x)^2+1}d(c+d x)+(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}\) |
| \(\Big \downarrow \) 5456 |
| \(\displaystyle \frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3+3 b \left (\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 b}-\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{-c-d x+i}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5380 |
| \(\displaystyle \frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3+3 b \left (-2 b \int \frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)+\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2\right )}{d}\) |
| \(\Big \downarrow \) 5530 |
| \(\displaystyle \frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3+3 b \left (-2 b \left (-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )\right )+\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2\right )}{d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3+3 b \left (-2 b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )\right )+\frac {i \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2\right )}{d}\) |
((c + d*x)*(a + b*ArcCot[c + d*x])^3 + 3*b*(((I/3)*(a + b*ArcCot[c + d*x]) ^3)/b - (a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))] - 2*b*((-1/2*I) *(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))] + (b*PolyLog[ 3, 1 - 2/(1 + I*(c + d*x))])/4)))/d
3.2.43.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( p/e) Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 ]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*((a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1))), x] - Simp[ 1/(c*d) Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] - Simp[b*p*(I/2) Int[(a + b*ArcCot[c*x])^(p - 1)*(PolyLog[2, 1 - u]/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c ^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - c*x)))^2, 0]
Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (136 ) = 272\).
Time = 2.46 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.76
| method | result | size |
| derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccot}\left (d x +c \right )^{3} \left (d x +c -i\right )+2 i \operatorname {arccot}\left (d x +c \right )^{3}-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c -i\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {arccot}\left (d x +c \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(395\) |
| default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccot}\left (d x +c \right )^{3} \left (d x +c -i\right )+2 i \operatorname {arccot}\left (d x +c \right )^{3}-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c -i\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {arccot}\left (d x +c \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(395\) |
| parts | \(a^{3} x +\frac {b^{3} \left (\operatorname {arccot}\left (d x +c \right )^{3} \left (d x +c -i\right )+2 i \operatorname {arccot}\left (d x +c \right )^{3}-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-3 \operatorname {arccot}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+6 i \operatorname {arccot}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{d}+\frac {3 a \,b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c -i\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )-2 \,\operatorname {arccot}\left (d x +c \right ) \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {arccot}\left (d x +c \right )^{2}+2 i \operatorname {polylog}\left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )+2 i \operatorname {polylog}\left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{d}+\frac {3 a^{2} b \left (\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(396\) |
1/d*((d*x+c)*a^3+b^3*(arccot(d*x+c)^3*(d*x+c-I)+2*I*arccot(d*x+c)^3-3*arcc ot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+6*I*arccot(d*x+c)*polylog( 2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/ 2))-3*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+6*I*arccot(d*x+c )*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*polylog(3,(d*x+c+I)/(1+(d*x+c )^2)^(1/2)))+3*a*b^2*(arccot(d*x+c)^2*(d*x+c-I)-2*arccot(d*x+c)*ln(1+(d*x+ c+I)/(1+(d*x+c)^2)^(1/2))-2*arccot(d*x+c)*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/ 2))+2*I*arccot(d*x+c)^2+2*I*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+2*I* polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2)))+3*a^2*b*(arccot(d*x+c)*(d*x+c)+1 /2*ln(1+(d*x+c)^2)))
\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(b^3*arccot(d*x + c)^3 + 3*a*b^2*arccot(d*x + c)^2 + 3*a^2*b*arcco t(d*x + c) + a^3, x)
\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3}\, dx \]
\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/8*b^3*x*arctan2(1, d*x + c)^3 - 3/32*b^3*x*arctan2(1, d*x + c)*log(d^2*x ^2 + 2*c*d*x + c^2 + 1)^2 + a^3*x + 3/2*(2*(d*x + c)*arccot(d*x + c) + log ((d*x + c)^2 + 1))*a^2*b/d + integrate(1/32*(28*b^3*arctan2(1, d*x + c)^3 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*x^2 + 96*a*b^2*arctan2(1, d*x + c)^2 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a* b^2*arctan2(1, d*x + c)^2)*c^2 + 4*(3*b^3*arctan2(1, d*x + c)^2 + 2*(7*b^3 *arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*x + 3*(b^3*d ^2*x^2*arctan2(1, d*x + c) + b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c) + (2*b^3*c*arctan2(1, d*x + c) - b^3)*d*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 12*(b^3*d^2*x^2*arctan2(1, d*x + c) + b^3*c*d*x*arctan2(1, d*x + c))*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)
\[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int {\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]