Integrand size = 12, antiderivative size = 143 \[ \int (a+b \arctan (c+d x))^3 \, dx=\frac {i (a+b \arctan (c+d x))^3}{d}+\frac {(c+d x) (a+b \arctan (c+d x))^3}{d}+\frac {3 b (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {3 i b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \]
I*(a+b*arctan(d*x+c))^3/d+(d*x+c)*(a+b*arctan(d*x+c))^3/d+3*b*(a+b*arctan( d*x+c))^2*ln(2/(1+I*(d*x+c)))/d+3*I*b^2*(a+b*arctan(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 = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.48 \[ \int (a+b \arctan (c+d x))^3 \, dx=\frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \arctan (c+d x)-3 a^2 b \log \left (1+(c+d x)^2\right )+6 a b^2 \left (\arctan (c+d x) \left ((-i+c+d x) \arctan (c+d x)+2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )+2 b^3 \left (\arctan (c+d x)^2 \left ((-i+c+d x) \arctan (c+d x)+3 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-3 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )}{2 d} \]
(2*a^3*(c + d*x) + 6*a^2*b*(c + d*x)*ArcTan[c + d*x] - 3*a^2*b*Log[1 + (c + d*x)^2] + 6*a*b^2*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*L og[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x ])]) + 2*b^3*(ArcTan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I) *ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2))/(2*d)
Time = 0.63 (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 = {5562, 5345, 5455, 5379, 5529, 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.
\(\displaystyle \int (a+b \arctan (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 5562 |
\(\displaystyle \frac {\int (a+b \arctan (c+d x))^3d(c+d x)}{d}\) |
\(\Big \downarrow \) 5345 |
\(\displaystyle \frac {(c+d x) (a+b \arctan (c+d x))^3-3 b \int \frac {(c+d x) (a+b \arctan (c+d x))^2}{(c+d x)^2+1}d(c+d x)}{d}\) |
\(\Big \downarrow \) 5455 |
\(\displaystyle \frac {(c+d x) (a+b \arctan (c+d x))^3-3 b \left (-\int \frac {(a+b \arctan (c+d x))^2}{-c-d x+i}d(c+d x)-\frac {i (a+b \arctan (c+d x))^3}{3 b}\right )}{d}\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle \frac {(c+d x) (a+b \arctan (c+d x))^3-3 b \left (2 b \int \frac {(a+b \arctan (c+d x)) \log \left (\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)-\frac {i (a+b \arctan (c+d x))^3}{3 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2\right )}{d}\) |
\(\Big \downarrow \) 5529 |
\(\displaystyle \frac {(c+d x) (a+b \arctan (c+d x))^3-3 b \left (2 b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))\right )-\frac {i (a+b \arctan (c+d x))^3}{3 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2\right )}{d}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {(c+d x) (a+b \arctan (c+d x))^3-3 b \left (2 b \left (-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )\right )-\frac {i (a+b \arctan (c+d x))^3}{3 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2\right )}{d}\) |
((c + d*x)*(a + b*ArcTan[c + d*x])^3 - 3*b*(((-1/3*I)*(a + b*ArcTan[c + d* x])^3)/b - (a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))] + 2*b*((-1/2 *I)*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))] - (b*PolyL og[3, 1 - 2/(1 + I*(c + d*x))])/4)))/d
3.1.38.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*p*(I/2) Int[(a + b*ArcTan[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_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcTan[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. 278 vs. \(2 (136 ) = 272\).
Time = 0.78 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.95
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arctan \left (d x +c \right )^{3} \left (d x +c +i\right )-2 i \arctan \left (d x +c \right )^{3}+3 \arctan \left (d x +c \right )^{2} \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-3 i \arctan \left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )}{2}\right )+3 a \,b^{2} \left (\arctan \left (d x +c \right )^{2} \left (d x +c +i\right )+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-2 i \arctan \left (d x +c \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )\right )+3 a^{2} b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(279\) |
default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arctan \left (d x +c \right )^{3} \left (d x +c +i\right )-2 i \arctan \left (d x +c \right )^{3}+3 \arctan \left (d x +c \right )^{2} \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-3 i \arctan \left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )}{2}\right )+3 a \,b^{2} \left (\arctan \left (d x +c \right )^{2} \left (d x +c +i\right )+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-2 i \arctan \left (d x +c \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )\right )+3 a^{2} b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(279\) |
parts | \(a^{3} x +\frac {b^{3} \left (\arctan \left (d x +c \right )^{3} \left (d x +c +i\right )-2 i \arctan \left (d x +c \right )^{3}+3 \arctan \left (d x +c \right )^{2} \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-3 i \arctan \left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )}{2}\right )}{d}+\frac {3 a^{2} b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (\arctan \left (d x +c \right )^{2} \left (d x +c +i\right )+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-2 i \arctan \left (d x +c \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )\right )}{d}\) | \(280\) |
1/d*((d*x+c)*a^3+b^3*(arctan(d*x+c)^3*(d*x+c+I)-2*I*arctan(d*x+c)^3+3*arct an(d*x+c)^2*ln(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))-3*I*arctan(d*x+c)*polylog( 2,-(1+I*(d*x+c))^2/(1+(d*x+c)^2))+3/2*polylog(3,-(1+I*(d*x+c))^2/(1+(d*x+c )^2)))+3*a*b^2*(arctan(d*x+c)^2*(d*x+c+I)+2*arctan(d*x+c)*ln(1+(1+I*(d*x+c ))^2/(1+(d*x+c)^2))-2*I*arctan(d*x+c)^2-I*polylog(2,-(1+I*(d*x+c))^2/(1+(d *x+c)^2)))+3*a^2*b*((d*x+c)*arctan(d*x+c)-1/2*ln(1+(d*x+c)^2)))
\[ \int (a+b \arctan (c+d x))^3 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arcta n(d*x + c) + a^3, x)
\[ \int (a+b \arctan (c+d x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{3}\, dx \]
\[ \int (a+b \arctan (c+d x))^3 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
7/8*b^3*c^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 1/8*b^3*x*arctan (d*x + c)^3 + 3*a*b^2*c^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 3/ 32*b^3*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 - (3*arctan(d* x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*c^ 2 - 7/32*(6*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((d^2*x + c*d)/d)^4/d)*b^3*c^2 + 7 /8*b^3*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 28*b^3*d^2*integrate( 1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*i ntegrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2* x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*d^2*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c*d*integrate(1/32*x*arc tan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*d^2*integrate(1/ 32*x^2*arctan(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) + 6*b^3*c*d*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c*d*in tegrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^ 3*c*d*integrate(1/32*x*arctan(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) + 3*b^3*c^2*integrate(1/32*arctan(d*x + c) *log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3* a*b^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 12*b^3*d*integrate(...
\[ \int (a+b \arctan (c+d x))^3 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (a+b \arctan (c+d x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \]