Integrand size = 14, antiderivative size = 255 \[ \int x^5 (a+b \arctan (c x))^3 \, dx=\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \arctan (c x)}{60 c^6}-\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {23 b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {23 i b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{30 c^6} \]
19/60*b^3*x/c^5-1/60*b^3*x^3/c^3-19/60*b^3*arctan(c*x)/c^6-4/15*b^2*x^2*(a +b*arctan(c*x))/c^4+1/20*b^2*x^4*(a+b*arctan(c*x))/c^2-23/30*I*b*(a+b*arct an(c*x))^2/c^6-1/2*b*x*(a+b*arctan(c*x))^2/c^5+1/6*b*x^3*(a+b*arctan(c*x)) ^2/c^3-1/10*b*x^5*(a+b*arctan(c*x))^2/c+1/6*(a+b*arctan(c*x))^3/c^6+1/6*x^ 6*(a+b*arctan(c*x))^3-23/15*b^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^6-23/3 0*I*b^3*polylog(2,1-2/(1+I*c*x))/c^6
Time = 0.95 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b \arctan (c x))^3 \, dx=\frac {-19 a b^2-30 a^2 b c x+19 b^3 c x-16 a b^2 c^2 x^2+10 a^2 b c^3 x^3-b^3 c^3 x^3+3 a b^2 c^4 x^4-6 a^2 b c^5 x^5+10 a^3 c^6 x^6+2 b^2 \left (b \left (23 i-15 c x+5 c^3 x^3-3 c^5 x^5\right )+15 a \left (1+c^6 x^6\right )\right ) \arctan (c x)^2+10 b^3 \left (1+c^6 x^6\right ) \arctan (c x)^3+b \arctan (c x) \left (b^2 \left (-19-16 c^2 x^2+3 c^4 x^4\right )-4 a b c x \left (15-5 c^2 x^2+3 c^4 x^4\right )+30 a^2 \left (1+c^6 x^6\right )-92 b^2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+46 a b^2 \log \left (1+c^2 x^2\right )+46 i b^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{60 c^6} \]
(-19*a*b^2 - 30*a^2*b*c*x + 19*b^3*c*x - 16*a*b^2*c^2*x^2 + 10*a^2*b*c^3*x ^3 - b^3*c^3*x^3 + 3*a*b^2*c^4*x^4 - 6*a^2*b*c^5*x^5 + 10*a^3*c^6*x^6 + 2* b^2*(b*(23*I - 15*c*x + 5*c^3*x^3 - 3*c^5*x^5) + 15*a*(1 + c^6*x^6))*ArcTa n[c*x]^2 + 10*b^3*(1 + c^6*x^6)*ArcTan[c*x]^3 + b*ArcTan[c*x]*(b^2*(-19 - 16*c^2*x^2 + 3*c^4*x^4) - 4*a*b*c*x*(15 - 5*c^2*x^2 + 3*c^4*x^4) + 30*a^2* (1 + c^6*x^6) - 92*b^2*Log[1 + E^((2*I)*ArcTan[c*x])]) + 46*a*b^2*Log[1 + c^2*x^2] + (46*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(60*c^6)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(525\) vs. \(2(255)=510\).
Time = 2.90 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.06, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {5361, 5451, 5361, 5451, 5361, 254, 2009, 5451, 5345, 5361, 262, 216, 5419, 5455, 5379, 2849, 2752}
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.
| \(\displaystyle \int x^5 (a+b \arctan (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \int \frac {x^6 (a+b \arctan (c x))^2}{c^2 x^2+1}dx\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\int x^4 (a+b \arctan (c x))^2dx}{c^2}-\frac {\int \frac {x^4 (a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x^4 (a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\int x^3 (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {\int x^2 (a+b \arctan (c x))^2dx}{c^2}-\frac {\int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \int \frac {x^4}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \int \left (\frac {x^2}{c^2}+\frac {1}{c^4 \left (c^2 x^2+1\right )}-\frac {1}{c^4}\right )dx}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {\int (a+b \arctan (c x))^2dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \int \frac {x^2}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \int \frac {x^2}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\int \frac {1}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\int \frac {1}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {(a+b \arctan (c x))^3}{3 b c^3}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )}{c^2}-\frac {-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )}{c^2}-\frac {-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )}{c^2}-\frac {-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )}{c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} b c \left (\frac {\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}}{c^2}\right )}{c^2}-\frac {\frac {\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}\right )}{c^2}-\frac {-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )}{c^2}}{c^2}}{c^2}\right )\) |
(x^6*(a + b*ArcTan[c*x])^3)/6 - (b*c*(((x^5*(a + b*ArcTan[c*x])^2)/5 - (2* b*c*(((x^4*(a + b*ArcTan[c*x]))/4 - (b*c*(-(x/c^4) + x^3/(3*c^2) + ArcTan[ c*x]/c^5))/4)/c^2 - (((x^2*(a + b*ArcTan[c*x]))/2 - (b*c*(x/c^2 - ArcTan[c *x]/c^3))/2)/c^2 - (((-1/2*I)*(a + b*ArcTan[c*x])^2)/(b*c^2) - (((a + b*Ar cTan[c*x])*Log[2/(1 + I*c*x)])/c + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)]) /c)/c)/c^2)/c^2))/5)/c^2 - (((x^3*(a + b*ArcTan[c*x])^2)/3 - (2*b*c*(((x^2 *(a + b*ArcTan[c*x]))/2 - (b*c*(x/c^2 - ArcTan[c*x]/c^3))/2)/c^2 - (((-1/2 *I)*(a + b*ArcTan[c*x])^2)/(b*c^2) - (((a + b*ArcTan[c*x])*Log[2/(1 + I*c* x)])/c + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/c)/c)/c^2))/3)/c^2 - (-1/ 3*(a + b*ArcTan[c*x])^3/(b*c^3) + (x*(a + b*ArcTan[c*x])^2 - 2*b*c*(((-1/2 *I)*(a + b*ArcTan[c*x])^2)/(b*c^2) - (((a + b*ArcTan[c*x])*Log[2/(1 + I*c* x)])/c + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/c)/c))/c^2)/c^2)/c^2))/2
3.1.24.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -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_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
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]
Time = 2.52 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.58
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} c^{6} x^{6}}{6}+b^{3} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{3}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{10}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{6}-\frac {\arctan \left (c x \right )^{2} c x}{2}+\frac {\arctan \left (c x \right )^{3}}{6}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{20}-\frac {4 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {23 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {c^{3} x^{3}}{60}+\frac {19 c x}{60}-\frac {19 \arctan \left (c x \right )}{60}+\frac {23 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{60}-\frac {23 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{60}\right )+3 a \,b^{2} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{9}-\frac {c x \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right )^{2}}{6}+\frac {c^{4} x^{4}}{60}-\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c^{2} x^{2}+1\right )}{90}\right )+3 a^{2} b \left (\frac {c^{6} x^{6} \arctan \left (c x \right )}{6}-\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}-\frac {c x}{6}+\frac {\arctan \left (c x \right )}{6}\right )}{c^{6}}\) | \(402\) |
| default | \(\frac {\frac {a^{3} c^{6} x^{6}}{6}+b^{3} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{3}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{10}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{6}-\frac {\arctan \left (c x \right )^{2} c x}{2}+\frac {\arctan \left (c x \right )^{3}}{6}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{20}-\frac {4 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {23 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {c^{3} x^{3}}{60}+\frac {19 c x}{60}-\frac {19 \arctan \left (c x \right )}{60}+\frac {23 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{60}-\frac {23 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{60}\right )+3 a \,b^{2} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{9}-\frac {c x \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right )^{2}}{6}+\frac {c^{4} x^{4}}{60}-\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c^{2} x^{2}+1\right )}{90}\right )+3 a^{2} b \left (\frac {c^{6} x^{6} \arctan \left (c x \right )}{6}-\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}-\frac {c x}{6}+\frac {\arctan \left (c x \right )}{6}\right )}{c^{6}}\) | \(402\) |
| parts | \(\frac {a^{3} x^{6}}{6}+\frac {b^{3} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{3}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{10}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{6}-\frac {\arctan \left (c x \right )^{2} c x}{2}+\frac {\arctan \left (c x \right )^{3}}{6}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{20}-\frac {4 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {23 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {c^{3} x^{3}}{60}+\frac {19 c x}{60}-\frac {19 \arctan \left (c x \right )}{60}+\frac {23 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{60}-\frac {23 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{60}\right )}{c^{6}}+\frac {3 a \,b^{2} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{9}-\frac {c x \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right )^{2}}{6}+\frac {c^{4} x^{4}}{60}-\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c^{2} x^{2}+1\right )}{90}\right )}{c^{6}}+\frac {3 a^{2} b \left (\frac {c^{6} x^{6} \arctan \left (c x \right )}{6}-\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}-\frac {c x}{6}+\frac {\arctan \left (c x \right )}{6}\right )}{c^{6}}\) | \(404\) |
| risch | \(\text {Expression too large to display}\) | \(1277\) |
1/c^6*(1/6*a^3*c^6*x^6+b^3*(1/6*c^6*x^6*arctan(c*x)^3-1/10*c^5*x^5*arctan( c*x)^2+1/6*c^3*x^3*arctan(c*x)^2-1/2*arctan(c*x)^2*c*x+1/6*arctan(c*x)^3+1 /20*c^4*x^4*arctan(c*x)-4/15*c^2*x^2*arctan(c*x)+23/30*arctan(c*x)*ln(c^2* x^2+1)-1/60*c^3*x^3+19/60*c*x-19/60*arctan(c*x)+23/60*I*(ln(c*x-I)*ln(c^2* x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))) -23/60*I*(ln(c*x+I)*ln(c^2*x^2+1)-1/2*ln(c*x+I)^2-dilog(1/2*I*(c*x-I))-ln( c*x+I)*ln(1/2*I*(c*x-I))))+3*a*b^2*(1/6*c^6*x^6*arctan(c*x)^2-1/15*c^5*x^5 *arctan(c*x)+1/9*c^3*x^3*arctan(c*x)-1/3*c*x*arctan(c*x)+1/6*arctan(c*x)^2 +1/60*c^4*x^4-4/45*c^2*x^2+23/90*ln(c^2*x^2+1))+3*a^2*b*(1/6*c^6*x^6*arcta n(c*x)-1/30*c^5*x^5+1/18*c^3*x^3-1/6*c*x+1/6*arctan(c*x)))
\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5} \,d x } \]
integral(b^3*x^5*arctan(c*x)^3 + 3*a*b^2*x^5*arctan(c*x)^2 + 3*a^2*b*x^5*a rctan(c*x) + a^3*x^5, x)
\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int x^{5} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, dx \]
\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5} \,d x } \]
1/2*a*b^2*x^6*arctan(c*x)^2 + 1/6*a^3*x^6 + 1/30*(15*x^6*arctan(c*x) - c*( (3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*a^2*b - 1/60*(4* c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7)*arctan(c*x) - (3*c^4*x^4 - 16*c^2*x^2 - 30*arctan(c*x)^2 + 46*log(c^2*x^2 + 1))/c^6)*a*b ^2 + 1/480*(20*(5760*c^7*integrate(1/480*x^7*arctan(c*x)^3/(c^7*x^2 + c^5) , x) - 1440*c^6*integrate(1/480*x^6*arctan(c*x)^2/(c^7*x^2 + c^5), x) - 36 0*c^6*integrate(1/480*x^6*log(c^2*x^2 + 1)^2/(c^7*x^2 + c^5), x) - 288*c^6 *integrate(1/480*x^6*log(c^2*x^2 + 1)/(c^7*x^2 + c^5), x) + 5760*c^5*integ rate(1/480*x^5*arctan(c*x)^3/(c^7*x^2 + c^5), x) + 576*c^5*integrate(1/480 *x^5*arctan(c*x)/(c^7*x^2 + c^5), x) + 480*c^4*integrate(1/480*x^4*log(c^2 *x^2 + 1)/(c^7*x^2 + c^5), x) - 960*c^3*integrate(1/480*x^3*arctan(c*x)/(c ^7*x^2 + c^5), x) - 1440*c^2*integrate(1/480*x^2*log(c^2*x^2 + 1)/(c^7*x^2 + c^5), x) + 2880*c*integrate(1/480*x*arctan(c*x)/(c^7*x^2 + c^5), x) - a rctan(c*x)^3/c^6 - 360*integrate(1/480*log(c^2*x^2 + 1)^2/(c^7*x^2 + c^5), x))*c^6 + 40*(c^6*x^6 + 1)*arctan(c*x)^3 - 4*(3*c^5*x^5 - 5*c^3*x^3 + 15* c*x)*arctan(c*x)^2 + (3*c^5*x^5 - 5*c^3*x^3 + 15*c*x)*log(c^2*x^2 + 1)^2)* b^3/c^6
\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5} \,d x } \]
Timed out. \[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int x^5\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \]