Integrand size = 18, antiderivative size = 499 \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\frac {i c (a+b \arctan (c x))^3}{c^2 d^2+e^2}+\frac {c^2 d (a+b \arctan (c x))^3}{e \left (c^2 d^2+e^2\right )}-\frac {(a+b \arctan (c x))^3}{e (d+e x)}-\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )} \]
I*c*(a+b*arctan(c*x))^3/(c^2*d^2+e^2)+c^2*d*(a+b*arctan(c*x))^3/e/(c^2*d^2 +e^2)-(a+b*arctan(c*x))^3/e/(e*x+d)-3*b*c*(a+b*arctan(c*x))^2*ln(2/(1-I*c* x))/(c^2*d^2+e^2)+3*b*c*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/(c^2*d^2+e^2)+ 3*b*c*(a+b*arctan(c*x))^2*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2 )+3*I*b^2*c*(a+b*arctan(c*x))*polylog(2,1-2/(1-I*c*x))/(c^2*d^2+e^2)+3*I*b ^2*c*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/(c^2*d^2+e^2)-3*I*b^2*c*(a +b*arctan(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2) -3/2*b^3*c*polylog(3,1-2/(1-I*c*x))/(c^2*d^2+e^2)+3/2*b^3*c*polylog(3,1-2/ (1+I*c*x))/(c^2*d^2+e^2)+3/2*b^3*c*polylog(3,1-2*c*(e*x+d)/(c*d+I*e)/(1-I* c*x))/(c^2*d^2+e^2)
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx \]
Time = 0.78 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5389, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {3 b c \int \left (\frac {e^2 (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {c^2 (d-e x) (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right ) \left (c^2 x^2+1\right )}\right )dx}{e}-\frac {(a+b \arctan (c x))^3}{e (d+e x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^3}{e (d+e x)}+\frac {3 b c \left (\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c^2 d^2+e^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d^2+e^2}-\frac {i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac {c d (a+b \arctan (c x))^3}{3 b \left (c^2 d^2+e^2\right )}+\frac {i e (a+b \arctan (c x))^3}{3 b \left (c^2 d^2+e^2\right )}-\frac {e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2+e^2}+\frac {e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2+e^2}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}\right )}{e}\) |
-((a + b*ArcTan[c*x])^3/(e*(d + e*x))) + (3*b*c*((c*d*(a + b*ArcTan[c*x])^ 3)/(3*b*(c^2*d^2 + e^2)) + ((I/3)*e*(a + b*ArcTan[c*x])^3)/(b*(c^2*d^2 + e ^2)) - (e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (e*( a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2) + (e*(a + b*ArcTa n[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2) + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2 ) + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e ^2) - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I* e)*(1 - I*c*x))])/(c^2*d^2 + e^2) - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/ (2*(c^2*d^2 + e^2)) + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*(c^2*d^2 + e^2)) + (b^2*e*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/ (2*(c^2*d^2 + e^2))))/e
3.1.19.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 22.85 (sec) , antiderivative size = 2398, normalized size of antiderivative = 4.81
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2398\) |
| default | \(\text {Expression too large to display}\) | \(2398\) |
| parts | \(\text {Expression too large to display}\) | \(2406\) |
1/c*(-a^3*c^2/(c*e*x+c*d)/e+b^3*c^2*(-1/(c*e*x+c*d)/e*arctan(c*x)^3+3/e*(a rctan(c*x)^2*e/(c^2*d^2+e^2)*ln(c*e*x+c*d)-1/2*arctan(c*x)^2/(c^2*d^2+e^2) *e*ln(c^2*x^2+1)+1/3*arctan(c*x)^3/(c^2*d^2+e^2)*d*c+e/(c^2*d^2+e^2)*arcta n(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-e/(c^2*d^2+e^2)*arctan(c*x)^2*ln( -I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)-I/(c^2*d ^2+e^2)*e*c*d/(c*d-I*e)*arctan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x )^2/(c^2*x^2+1))+1/(c^2*d^2+e^2)*e*c*d/(c*d-I*e)*arctan(c*x)^2*ln(1-(I*e-c *d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2/(c^2*d^2+e^2)*e*c*d/(c*d-I*e)*p olylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-I*e^2*arctan(c*x)*po lylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d* c)+e^2*arctan(c*x)^2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^ 2*d^2+e^2)/(e+I*d*c)+1/2*e^2*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^ 2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)-1/3*I*e/(c^2*d^2+e^2)*arctan(c*x)^3+1/4* e/(c^2*d^2+e^2)*(I*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/( c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2+2*I*Pi*csgn(I*(-I*e*(1+I*c*x)^ 2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2 +1)))^3+I*Pi*csgn(I*(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))^2)+I*Pi*csgn(I/(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+I*c*x)^2/(c^2*x^2+1))^2)^2-I*Pi*csgn(I*(1+I*c*x)/(c^2 *x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))-2*I*Pi*csgn(I*(1+(1+I*...
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
integral((b^3*arctan(c*x)^3 + 3*a*b^2*arctan(c*x)^2 + 3*a^2*b*arctan(c*x) + a^3)/(e^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
3/2*((2*c*d*arctan(c*x)/(c^2*d^2*e + e^3) - log(c^2*x^2 + 1)/(c^2*d^2 + e^ 2) + 2*log(e*x + d)/(c^2*d^2 + e^2))*c - 2*arctan(c*x)/(e^2*x + d*e))*a^2* b - a^3/(e^2*x + d*e) - 1/32*(4*b^3*arctan(c*x)^3 - 3*b^3*arctan(c*x)*log( c^2*x^2 + 1)^2 - 32*(e^2*x + d*e)*integrate(1/32*(28*(b^3*c^2*e*x^2 + b^3* e)*arctan(c*x)^3 + 12*(8*a*b^2*c^2*e*x^2 + b^3*c*e*x + b^3*c*d + 8*a*b^2*e )*arctan(c*x)^2 - 12*(b^3*c^2*e*x^2 + b^3*c^2*d*x)*arctan(c*x)*log(c^2*x^2 + 1) - 3*(b^3*c*e*x + b^3*c*d - (b^3*c^2*e*x^2 + b^3*e)*arctan(c*x))*log( c^2*x^2 + 1)^2)/(c^2*e^3*x^4 + 2*c^2*d*e^2*x^3 + 2*d*e^2*x + d^2*e + (c^2* d^2*e + e^3)*x^2), x))/(e^2*x + d*e)
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^2} \,d x \]