Integrand size = 14, antiderivative size = 206 \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=2 (a+b \arctan (c x))^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )+\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right ) \]
-2*(a+b*arctan(c*x))^3*arctanh(-1+2/(1+I*c*x))-3/2*I*b*(a+b*arctan(c*x))^2 *polylog(2,1-2/(1+I*c*x))+3/2*I*b*(a+b*arctan(c*x))^2*polylog(2,-1+2/(1+I* c*x))-3/2*b^2*(a+b*arctan(c*x))*polylog(3,1-2/(1+I*c*x))+3/2*b^2*(a+b*arct an(c*x))*polylog(3,-1+2/(1+I*c*x))+3/4*I*b^3*polylog(4,1-2/(1+I*c*x))-3/4* I*b^3*polylog(4,-1+2/(1+I*c*x))
Time = 0.32 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=a^3 \log (c x)+\frac {3}{2} i a^2 b (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+3 a b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} i \arctan (c x)^3+\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )-\frac {1}{64} i b^3 \left (\pi ^4-32 \arctan (c x)^4+64 i \arctan (c x)^3 \log \left (1-e^{-2 i \arctan (c x)}\right )-64 i \arctan (c x)^3 \log \left (1+e^{2 i \arctan (c x)}\right )-96 \arctan (c x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-96 \arctan (c x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+96 i \arctan (c x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-96 i \arctan (c x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c x)}\right )\right ) \]
a^3*Log[c*x] + ((3*I)/2)*a^2*b*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + 3*a*b^2*((-1/24*I)*Pi^3 + ((2*I)/3)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[ 2, -E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLo g[3, -E^((2*I)*ArcTan[c*x])]/2) - (I/64)*b^3*(Pi^4 - 32*ArcTan[c*x]^4 + (6 4*I)*ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] - (64*I)*ArcTan[c*x]^3* Log[1 + E^((2*I)*ArcTan[c*x])] - 96*ArcTan[c*x]^2*PolyLog[2, E^((-2*I)*Arc Tan[c*x])] - 96*ArcTan[c*x]^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (96*I)* ArcTan[c*x]*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (96*I)*ArcTan[c*x]*PolyLo g[3, -E^((2*I)*ArcTan[c*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[c*x])] + 48* PolyLog[4, -E^((2*I)*ArcTan[c*x])])
Time = 0.96 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5357, 5523, 5529, 5533, 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.
\(\displaystyle \int \frac {(a+b \arctan (c x))^3}{x} \, dx\) |
\(\Big \downarrow \) 5357 |
\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3-6 b c \int \frac {(a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\) |
\(\Big \downarrow \) 5523 |
\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3-6 b c \left (\frac {1}{2} \int \frac {(a+b \arctan (c x))^2 \log \left (2-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {1}{2} \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )\) |
\(\Big \downarrow \) 5529 |
\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3-6 b c \left (\frac {1}{2} \left (\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}-i b \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )+\frac {1}{2} \left (i b \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 c}\right )\right )\) |
\(\Big \downarrow \) 5533 |
\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3-6 b c \left (\frac {1}{2} \left (\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}-i b \left (\frac {i \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )\right )+\frac {1}{2} \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{2 c}-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{c^2 x^2+1}dx\right )-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 c}\right )\right )\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3-6 b c \left (\frac {1}{2} \left (\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}-i b \left (\frac {i \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )}{4 c}\right )\right )+\frac {1}{2} \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (4,\frac {2}{i c x+1}-1\right )}{4 c}\right )-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 c}\right )\right )\) |
2*(a + b*ArcTan[c*x])^3*ArcTanh[1 - 2/(1 + I*c*x)] - 6*b*c*((((I/2)*(a + b *ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - I*b*(((I/2)*(a + b*ArcT an[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/c + (b*PolyLog[4, 1 - 2/(1 + I*c*x )])/(4*c)))/2 + (((-1/2*I)*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I* c*x)])/c + I*b*(((I/2)*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)]) /c + (b*PolyLog[4, -1 + 2/(1 + I*c*x)])/(4*c)))/2)
3.1.30.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 + I*c*x)], x] - Simp[2*b*c*p Int[(a + b *ArcTan[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x _)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTan[c*x])^p/(d + e *x^2)), x], x] - Simp[1/2 Int[Log[1 - u]*((a + b*ArcTan[c*x])^p/(d + e*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^2, 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_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_. )*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/(2* c*d)), x] - Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[k + 1 , u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^2, 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.58 (sec) , antiderivative size = 2026, normalized size of antiderivative = 9.83
\[\text {Expression too large to display}\]
a^3*ln(c*x)+b^3*(ln(c*x)*arctan(c*x)^3-arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x ^2+1)-1)+arctan(c*x)^3*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2 *polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c*x)*polylog(3,(1+I*c*x)/ (c^2*x^2+1)^(1/2))+6*I*polylog(4,(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^ 3*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2*polylog(2,-(1+I*c*x) /(c^2*x^2+1)^(1/2))+6*arctan(c*x)*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+ 6*I*polylog(4,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*Pi*(csgn(I*((1+I*c*x)^2/ (c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)- 1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c *x)^2/(c^2*x^2+1)))^2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I/(1+(1+I*c *x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2 *x^2+1)))-csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2 +1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2-csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))* csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2+csgn(I*( (1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3-csgn(I*((1+I*c*x )^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2 +1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2+csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+ (1+I*c*x)^2/(c^2*x^2+1)))^3+1)*arctan(c*x)^3+3/2*I*arctan(c*x)^2*polylog(2 ,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*arctan(c*x)*polylog(3,-(1+I*c*x)^2/(c^2*x^2 +1))-3/4*I*polylog(4,-(1+I*c*x)^2/(c^2*x^2+1)))+3*a*b^2*(ln(c*x)*arctan...
\[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
\[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}}{x}\, dx \]
\[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
a^3*log(x) + 1/32*integrate((28*b^3*arctan(c*x)^3 + 3*b^3*arctan(c*x)*log( c^2*x^2 + 1)^2 + 96*a*b^2*arctan(c*x)^2 + 96*a^2*b*arctan(c*x))/x, x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x} \,d x \]