Integrand size = 25, antiderivative size = 299 \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\frac {b^2}{16 d^3 (i-c x)^2}-\frac {11 i b^2}{16 d^3 (i-c x)}+\frac {11 i b^2 \arctan (c x)}{16 d^3}+\frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3} \]
1/16*b^2/d^3/(I-c*x)^2-11/16*I*b^2/d^3/(I-c*x)+11/16*I*b^2*arctan(c*x)/d^3 +1/4*I*b*(a+b*arctan(c*x))/d^3/(I-c*x)^2+5/4*b*(a+b*arctan(c*x))/d^3/(I-c* x)-5/8*(a+b*arctan(c*x))^2/d^3-1/2*(a+b*arctan(c*x))^2/d^3/(I-c*x)^2+I*(a+ b*arctan(c*x))^2/d^3/(I-c*x)-2*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x)) /d^3+(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/d^3+I*b*(a+b*arctan(c*x))*polylog (2,-1+2/(1+I*c*x))/d^3+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d^3
Time = 1.29 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\frac {-\frac {96 a^2}{(-i+c x)^2}-\frac {192 i a^2}{-i+c x}-192 i a^2 \arctan (c x)+192 a^2 \log (c x)-96 a^2 \log \left (1+c^2 x^2\right )+12 i a b \left (-32 \arctan (c x)^2-12 \cos (2 \arctan (c x))-\cos (4 \arctan (c x))-16 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+12 i \sin (2 \arctan (c x))-4 i \arctan (c x) \left (6 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+8 \log \left (1-e^{2 i \arctan (c x)}\right )-6 i \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))\right )+i \sin (4 \arctan (c x))\right )+b^2 \left (-8 i \pi ^3-72 \cos (2 \arctan (c x))-144 i \arctan (c x) \cos (2 \arctan (c x))+144 \arctan (c x)^2 \cos (2 \arctan (c x))-3 \cos (4 \arctan (c x))-12 i \arctan (c x) \cos (4 \arctan (c x))+24 \arctan (c x)^2 \cos (4 \arctan (c x))+192 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+192 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+96 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+72 i \sin (2 \arctan (c x))-144 \arctan (c x) \sin (2 \arctan (c x))-144 i \arctan (c x)^2 \sin (2 \arctan (c x))+3 i \sin (4 \arctan (c x))-12 \arctan (c x) \sin (4 \arctan (c x))-24 i \arctan (c x)^2 \sin (4 \arctan (c x))\right )}{192 d^3} \]
((-96*a^2)/(-I + c*x)^2 - ((192*I)*a^2)/(-I + c*x) - (192*I)*a^2*ArcTan[c* x] + 192*a^2*Log[c*x] - 96*a^2*Log[1 + c^2*x^2] + (12*I)*a*b*(-32*ArcTan[c *x]^2 - 12*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*PolyLog[2, E^((2*I )*ArcTan[c*x])] + (12*I)*Sin[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]*(6*Cos[2*A rcTan[c*x]] + Cos[4*ArcTan[c*x]] + 8*Log[1 - E^((2*I)*ArcTan[c*x])] - (6*I )*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) + I*Sin[4*ArcTan[c*x]]) + b^2 *((-8*I)*Pi^3 - 72*Cos[2*ArcTan[c*x]] - (144*I)*ArcTan[c*x]*Cos[2*ArcTan[c *x]] + 144*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] - 3*Cos[4*ArcTan[c*x]] - (12*I )*ArcTan[c*x]*Cos[4*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] + 1 92*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (192*I)*ArcTan[c*x]*Pol yLog[2, E^((-2*I)*ArcTan[c*x])] + 96*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (72*I)*Sin[2*ArcTan[c*x]] - 144*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (144*I)*A rcTan[c*x]^2*Sin[2*ArcTan[c*x]] + (3*I)*Sin[4*ArcTan[c*x]] - 12*ArcTan[c*x ]*Sin[4*ArcTan[c*x]] - (24*I)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(192*d^3)
Time = 0.91 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5411, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {(a+b \arctan (c x))^2}{d^3 x}-\frac {c (a+b \arctan (c x))^2}{d^3 (c x-i)}+\frac {i c (a+b \arctan (c x))^2}{d^3 (c x-i)^2}+\frac {c (a+b \arctan (c x))^2}{d^3 (c x-i)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^3}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (-c x+i)}+\frac {i b (a+b \arctan (c x))}{4 d^3 (-c x+i)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (-c x+i)}-\frac {(a+b \arctan (c x))^2}{2 d^3 (-c x+i)^2}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {11 i b^2 \arctan (c x)}{16 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^3}-\frac {11 i b^2}{16 d^3 (-c x+i)}+\frac {b^2}{16 d^3 (-c x+i)^2}\) |
b^2/(16*d^3*(I - c*x)^2) - (((11*I)/16)*b^2)/(d^3*(I - c*x)) + (((11*I)/16 )*b^2*ArcTan[c*x])/d^3 + ((I/4)*b*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (5*b*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)) - (5*(a + b*ArcTan[c*x])^2)/( 8*d^3) - (a + b*ArcTan[c*x])^2/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x] )^2)/(d^3*(I - c*x)) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] )/d^3 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (I*b*(a + b*ArcTa n[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^3)
3.2.16.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.61 (sec) , antiderivative size = 1783, normalized size of antiderivative = 5.96
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1783\) |
| default | \(\text {Expression too large to display}\) | \(1783\) |
| parts | \(\text {Expression too large to display}\) | \(1783\) |
a^2/d^3*ln(c*x)-1/2*a^2/d^3/(c*x-I)^2-I*a^2/d^3/(c*x-I)-1/2*a^2/d^3*ln(c^2 *x^2+1)-I*a^2/d^3*arctan(c*x)+b^2/d^3*(2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^( 1/2))+2*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-5/8*arctan(c*x)^2-arctan(c *x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^2*ln(c*x)+arctan(c*x)^2*ln (1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^( 1/2))-2*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c* x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*Pi*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))*csgn(I*((1+I*c*x)^2/(c^2 *x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^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)))*arctan(c*x)^2-1/2*I*Pi*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*arctan(c*x)^2-1/2*I*Pi*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*arcta n(c*x)^2+1/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2 +1)))^3*arctan(c*x)^2-1/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c* x)^2/(c^2*x^2+1)))^2*arctan(c*x)^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)))^3*arctan(c*x)^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)))^2*arctan(c*x)^2+arctan(c*x)^2*ln(2...
\[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{3} x} \,d x } \]
-1/8*(2*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*log(2*c*x/(c*x - I))*log(-(c*x + I)/(c*x - I))^2 + 4*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I)) - (2*I*b^2*c*x + 3*b^2)*log(-(c*x + I)/ (c*x - I))^2 - 8*(c^2*d^3*x^2 - 2*I*c*d^3*x - d^3)*integral(1/2*(2*I*a^2*c *x - 2*a^2 - (2*b^2*c^2*x^2 + (2*a*b - 3*I*b^2)*c*x + 2*I*a*b)*log(-(c*x + I)/(c*x - I)))/(c^4*d^3*x^5 - 2*I*c^3*d^3*x^4 - 2*I*c*d^3*x^2 - d^3*x), x ) - 4*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*polylog(3, -(c*x + I)/(c*x - I)))/ (c^2*d^3*x^2 - 2*I*c*d^3*x - d^3)
Exception generated. \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\text {Exception raised: RecursionError} \]
\[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{3} x} \,d x } \]
-1/128*(-16*I*a^2*c^2*x^2*arctan2(1, c*x) - 32*a^2*c*x*(arctan2(1, c*x) - 4*I) + 32*(I*b^2*c^2*x^2 + 2*b^2*c*x - I*b^2)*arctan(c*x)^3 - 4*(b^2*c^2*x ^2 - 2*I*b^2*c*x - b^2)*log(c^2*x^2 + 1)^3 + 16*a^2*(I*arctan2(1, c*x) + 1 2) + 16*(2*I*b^2*c*x + 3*b^2)*arctan(c*x)^2 - 4*(2*I*b^2*c*x + 3*b^2 - 2*( I*b^2*c^2*x^2 + 2*b^2*c*x - I*b^2)*arctan(c*x))*log(c^2*x^2 + 1)^2 + 6*(b^ 2*c^4*d^3*x^2 - 2*I*b^2*c^3*d^3*x - b^2*c^2*d^3)*(((8*c^2*x^2 + 7)*c^2/(c^ 12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1 )/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3))*c^4 + 2*(2*c^2*x^2 + 1)*c^2*lo g(c^2*x^2 + 1)^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - c^2*(c^2/(c^10* d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + 2*log(c^2*x^2 + 1)/(c^8*d^3*x^4 + 2*c ^6*d^3*x^2 + c^4*d^3)) - 512*c^2*integrate(1/16*x^3*arctan(c*x)^2/(c^6*d^3 *x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) - 2*log(c^2*x^2 + 1)^2/(c^ 6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + 512*integrate(1/16*x*arctan(c*x)^2/ (c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) - 2*(b^2*c^2*d^3* x^2 - 2*I*b^2*c*d^3*x - b^2*d^3)*(c^4*(c^2/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + 2*log(c^2*x^2 + 1)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3)) - 512*c^2*integrate(1/16*x^2*arctan(c*x)^2/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3* c^2*d^3*x^3 + d^3*x), x) + 2*c^2*log(c^2*x^2 + 1)^2/(c^6*d^3*x^4 + 2*c^4*d ^3*x^2 + c^2*d^3) + 512*integrate(1/16*arctan(c*x)^2/(c^6*d^3*x^7 + 3*c^4* d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x) + 128*integrate(1/16*log(c^2*x^2 +...
\[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]