Integrand size = 14, antiderivative size = 198 \[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=-\frac {b^3 c^3}{4 x}-\frac {1}{4} b^3 c^4 \arctan (c x)-\frac {b^2 c^2 (a+b \arctan (c x))}{4 x^2}+i b c^4 (a+b \arctan (c x))^2-\frac {b c (a+b \arctan (c x))^2}{4 x^3}+\frac {3 b c^3 (a+b \arctan (c x))^2}{4 x}+\frac {1}{4} c^4 (a+b \arctan (c x))^3-\frac {(a+b \arctan (c x))^3}{4 x^4}-2 b^2 c^4 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )+i b^3 c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right ) \]
-1/4*b^3*c^3/x-1/4*b^3*c^4*arctan(c*x)-1/4*b^2*c^2*(a+b*arctan(c*x))/x^2+I *b*c^4*(a+b*arctan(c*x))^2-1/4*b*c*(a+b*arctan(c*x))^2/x^3+3/4*b*c^3*(a+b* arctan(c*x))^2/x+1/4*c^4*(a+b*arctan(c*x))^3-1/4*(a+b*arctan(c*x))^3/x^4-2 *b^2*c^4*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))+I*b^3*c^4*polylog(2,-1+2/(1-I *c*x))
Time = 0.86 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=-\frac {a^3+a^2 b c x+a b^2 c^2 x^2-3 a^2 b c^3 x^3+b^3 c^3 x^3+a b^2 c^4 x^4+b^2 \left (b c x \left (1-3 c^2 x^2-4 i c^3 x^3\right )+a \left (3-3 c^4 x^4\right )\right ) \arctan (c x)^2-b^3 \left (-1+c^4 x^4\right ) \arctan (c x)^3+b \arctan (c x) \left (b^2 c^2 x^2 \left (1+c^2 x^2\right )+a b \left (2 c x-6 c^3 x^3\right )+a^2 \left (3-3 c^4 x^4\right )+8 b^2 c^4 x^4 \log \left (1-e^{2 i \arctan (c x)}\right )\right )+8 a b^2 c^4 x^4 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-4 i b^3 c^4 x^4 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )}{4 x^4} \]
-1/4*(a^3 + a^2*b*c*x + a*b^2*c^2*x^2 - 3*a^2*b*c^3*x^3 + b^3*c^3*x^3 + a* b^2*c^4*x^4 + b^2*(b*c*x*(1 - 3*c^2*x^2 - (4*I)*c^3*x^3) + a*(3 - 3*c^4*x^ 4))*ArcTan[c*x]^2 - b^3*(-1 + c^4*x^4)*ArcTan[c*x]^3 + b*ArcTan[c*x]*(b^2* c^2*x^2*(1 + c^2*x^2) + a*b*(2*c*x - 6*c^3*x^3) + a^2*(3 - 3*c^4*x^4) + 8* b^2*c^4*x^4*Log[1 - E^((2*I)*ArcTan[c*x])]) + 8*a*b^2*c^4*x^4*Log[(c*x)/Sq rt[1 + c^2*x^2]] - (4*I)*b^3*c^4*x^4*PolyLog[2, E^((2*I)*ArcTan[c*x])])/x^ 4
Time = 1.54 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.40, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5361, 5453, 5361, 5453, 5361, 264, 216, 5419, 5459, 5403, 2897}
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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {3}{4} b c \int \frac {(a+b \arctan (c x))^2}{x^4 \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {3}{4} b c \left (\int \frac {(a+b \arctan (c x))^2}{x^4}dx-c^2 \int \frac {(a+b \arctan (c x))^2}{x^2 \left (c^2 x^2+1\right )}dx\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\int \frac {(a+b \arctan (c x))^2}{x^2 \left (c^2 x^2+1\right )}dx\right )+\frac {2}{3} b c \int \frac {a+b \arctan (c x)}{x^3 \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (\int \frac {(a+b \arctan (c x))^2}{x^2}dx-c^2 \int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )\right )+\frac {2}{3} b c \left (\int \frac {a+b \arctan (c x)}{x^3}dx-c^2 \int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (c^2 \left (-\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )+2 b c \int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x))^2}{x}\right )\right )+\frac {2}{3} b c \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx\right )+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx-\frac {a+b \arctan (c x)}{2 x^2}\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (c^2 \left (-\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )+2 b c \int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x))^2}{x}\right )\right )+\frac {2}{3} b c \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx\right )+\frac {1}{2} b c \left (c^2 \left (-\int \frac {1}{c^2 x^2+1}dx\right )-\frac {1}{x}\right )-\frac {a+b \arctan (c x)}{2 x^2}\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (c^2 \left (-\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )+2 b c \int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x))^2}{x}\right )\right )+\frac {2}{3} b c \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx\right )-\frac {a+b \arctan (c x)}{2 x^2}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {3}{4} b c \left (-\left (c^2 \left (2 b c \int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx-\frac {c (a+b \arctan (c x))^3}{3 b}-\frac {(a+b \arctan (c x))^2}{x}\right )\right )+\frac {2}{3} b c \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x \left (c^2 x^2+1\right )}dx\right )-\frac {a+b \arctan (c x)}{2 x^2}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )-\frac {(a+b \arctan (c x))^3}{4 x^4}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^3}{4 x^4}+\frac {3}{4} b c \left (\frac {2}{3} b c \left (-\left (c^2 \left (i \int \frac {a+b \arctan (c x)}{x (c x+i)}dx-\frac {i (a+b \arctan (c x))^2}{2 b}\right )\right )-\frac {a+b \arctan (c x)}{2 x^2}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )-\left (c^2 \left (2 b c \left (i \int \frac {a+b \arctan (c x)}{x (c x+i)}dx-\frac {i (a+b \arctan (c x))^2}{2 b}\right )-\frac {c (a+b \arctan (c x))^3}{3 b}-\frac {(a+b \arctan (c x))^2}{x}\right )\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^3}{4 x^4}+\frac {3}{4} b c \left (-\left (c^2 \left (2 b c \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{c^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))\right )-\frac {i (a+b \arctan (c x))^2}{2 b}\right )-\frac {c (a+b \arctan (c x))^3}{3 b}-\frac {(a+b \arctan (c x))^2}{x}\right )\right )+\frac {2}{3} b c \left (-\left (c^2 \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{c^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))\right )-\frac {i (a+b \arctan (c x))^2}{2 b}\right )\right )-\frac {a+b \arctan (c x)}{2 x^2}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^3}{4 x^4}+\frac {3}{4} b c \left (\frac {2}{3} b c \left (-\left (c^2 \left (i \left (-i \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )\right )-\frac {i (a+b \arctan (c x))^2}{2 b}\right )\right )-\frac {a+b \arctan (c x)}{2 x^2}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )-\left (c^2 \left (2 b c \left (i \left (-i \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )\right )-\frac {i (a+b \arctan (c x))^2}{2 b}\right )-\frac {c (a+b \arctan (c x))^3}{3 b}-\frac {(a+b \arctan (c x))^2}{x}\right )\right )-\frac {(a+b \arctan (c x))^2}{3 x^3}\right )\) |
-1/4*(a + b*ArcTan[c*x])^3/x^4 + (3*b*c*(-1/3*(a + b*ArcTan[c*x])^2/x^3 - c^2*(-((a + b*ArcTan[c*x])^2/x) - (c*(a + b*ArcTan[c*x])^3)/(3*b) + 2*b*c* (((-1/2*I)*(a + b*ArcTan[c*x])^2)/b + I*((-I)*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - (b*PolyLog[2, -1 + 2/(1 - I*c*x)])/2))) + (2*b*c*(-1/2*(a + b*ArcTan[c*x])/x^2 + (b*c*(-x^(-1) - c*ArcTan[c*x]))/2 - c^2*(((-1/2*I) *(a + b*ArcTan[c*x])^2)/b + I*((-I)*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c *x)] - (b*PolyLog[2, -1 + 2/(1 - I*c*x)])/2))))/3))/4
3.1.34.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si mp[b*c*(p/d) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) 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] && LtQ[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*d*(p + 1))), x] + Si mp[I/d Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (180 ) = 360\).
Time = 4.50 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.14
| method | result | size |
| derivativedivides | \(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\arctan \left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {\arctan \left (c x \right )^{3}}{4}-\frac {\arctan \left (c x \right )^{2}}{4 c^{3} x^{3}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c x}+\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\arctan \left (c x \right )}{4 c^{2} x^{2}}-2 \ln \left (c x \right ) \arctan \left (c x \right )-\frac {\arctan \left (c x \right )}{4}-\frac {1}{4 c x}-i \ln \left (c x \right ) \ln \left (i c x +1\right )+i \ln \left (c x \right ) \ln \left (-i c x +1\right )-i \operatorname {dilog}\left (i c x +1\right )+i \operatorname {dilog}\left (-i c x +1\right )+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )+3 a \,b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}+\frac {\arctan \left (c x \right )}{2 c x}+\frac {\arctan \left (c x \right )^{2}}{4}-\frac {1}{12 c^{2} x^{2}}-\frac {2 \ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{3}\right )+3 a^{2} b \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {1}{12 c^{3} x^{3}}+\frac {1}{4 c x}+\frac {\arctan \left (c x \right )}{4}\right )\right )\) | \(423\) |
| default | \(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\arctan \left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {\arctan \left (c x \right )^{3}}{4}-\frac {\arctan \left (c x \right )^{2}}{4 c^{3} x^{3}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c x}+\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\arctan \left (c x \right )}{4 c^{2} x^{2}}-2 \ln \left (c x \right ) \arctan \left (c x \right )-\frac {\arctan \left (c x \right )}{4}-\frac {1}{4 c x}-i \ln \left (c x \right ) \ln \left (i c x +1\right )+i \ln \left (c x \right ) \ln \left (-i c x +1\right )-i \operatorname {dilog}\left (i c x +1\right )+i \operatorname {dilog}\left (-i c x +1\right )+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )+3 a \,b^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}+\frac {\arctan \left (c x \right )}{2 c x}+\frac {\arctan \left (c x \right )^{2}}{4}-\frac {1}{12 c^{2} x^{2}}-\frac {2 \ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{3}\right )+3 a^{2} b \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {1}{12 c^{3} x^{3}}+\frac {1}{4 c x}+\frac {\arctan \left (c x \right )}{4}\right )\right )\) | \(423\) |
| parts | \(-\frac {a^{3}}{4 x^{4}}+b^{3} c^{4} \left (-\frac {\arctan \left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {\arctan \left (c x \right )^{3}}{4}-\frac {\arctan \left (c x \right )^{2}}{4 c^{3} x^{3}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c x}+\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\arctan \left (c x \right )}{4 c^{2} x^{2}}-2 \ln \left (c x \right ) \arctan \left (c x \right )-\frac {\arctan \left (c x \right )}{4}-\frac {1}{4 c x}-i \ln \left (c x \right ) \ln \left (i c x +1\right )+i \ln \left (c x \right ) \ln \left (-i c x +1\right )-i \operatorname {dilog}\left (i c x +1\right )+i \operatorname {dilog}\left (-i c x +1\right )+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )+3 a^{2} b \,c^{4} \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {1}{12 c^{3} x^{3}}+\frac {1}{4 c x}+\frac {\arctan \left (c x \right )}{4}\right )+3 a \,b^{2} c^{4} \left (-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}+\frac {\arctan \left (c x \right )}{2 c x}+\frac {\arctan \left (c x \right )^{2}}{4}-\frac {1}{12 c^{2} x^{2}}-\frac {2 \ln \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{3}\right )\) | \(425\) |
c^4*(-1/4*a^3/c^4/x^4+b^3*(-1/4/c^4/x^4*arctan(c*x)^3+1/4*arctan(c*x)^3-1/ 4/c^3/x^3*arctan(c*x)^2+3/4*arctan(c*x)^2/c/x+arctan(c*x)*ln(c^2*x^2+1)-1/ 4/c^2/x^2*arctan(c*x)-2*ln(c*x)*arctan(c*x)-1/4*arctan(c*x)-1/4/c/x-I*ln(c *x)*ln(1+I*c*x)+I*ln(c*x)*ln(1-I*c*x)-I*dilog(1+I*c*x)+I*dilog(1-I*c*x)+1/ 2*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)))-1/2*I*(ln(c*x+I)*ln(c^2*x^2+1)-1/2*ln(c*x+I)^2-dilo g(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))))+3*a*b^2*(-1/4/c^4/x^4*arcta n(c*x)^2-1/6/c^3/x^3*arctan(c*x)+1/2/c/x*arctan(c*x)+1/4*arctan(c*x)^2-1/1 2/c^2/x^2-2/3*ln(c*x)+1/3*ln(c^2*x^2+1))+3*a^2*b*(-1/4/c^4/x^4*arctan(c*x) -1/12/c^3/x^3+1/4/c/x+1/4*arctan(c*x)))
\[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x^{5}} \,d x } \]
\[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x^5} \,d x \]