Integrand size = 16, antiderivative size = 214 \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{4} b^3 c^3 x+\frac {1}{4} b^3 c^4 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-i b c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {3}{4} b c^3 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} b c x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+2 b^2 c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )-i b^3 c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \] Output:
1/4*b^3*c^3*x+1/4*b^3*c^4*arccot(x/c)+1/4*b^2*c^2*x^2*(a+b*arccot(x/c))-I* b*c^4*(a+b*arccot(x/c))^2-3/4*b*c^3*x*(a+b*arccot(x/c))^2+1/4*b*c*x^3*(a+b *arccot(x/c))^2-1/4*c^4*(a+b*arccot(x/c))^3+1/4*x^4*(a+b*arccot(x/c))^3+2* b^2*c^4*(a+b*arccot(x/c))*ln(2-2/(1-I*c/x))-I*b^3*c^4*polylog(2,-1+2/(1-I* c/x))
Time = 0.43 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.18 \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{4} \left (a b^2 c^4-3 a^2 b c^3 x+b^3 c^3 x+a b^2 c^2 x^2+a^2 b c x^3+a^3 x^4+b^2 \left (b c \left (-4 i c^3-3 c^2 x+x^3\right )+3 a \left (-c^4+x^4\right )\right ) \arctan \left (\frac {c}{x}\right )^2+b^3 \left (-c^4+x^4\right ) \arctan \left (\frac {c}{x}\right )^3+b \arctan \left (\frac {c}{x}\right ) \left (2 a b c x \left (-3 c^2+x^2\right )+b^2 c^2 \left (c^2+x^2\right )+3 a^2 \left (-c^4+x^4\right )+8 b^2 c^4 \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )+8 a b^2 c^4 \log \left (\frac {c}{\sqrt {1+\frac {c^2}{x^2}} x}\right )-4 i b^3 c^4 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right ) \] Input:
Integrate[x^3*(a + b*ArcTan[c/x])^3,x]
Output:
(a*b^2*c^4 - 3*a^2*b*c^3*x + b^3*c^3*x + a*b^2*c^2*x^2 + a^2*b*c*x^3 + a^3 *x^4 + b^2*(b*c*((-4*I)*c^3 - 3*c^2*x + x^3) + 3*a*(-c^4 + x^4))*ArcTan[c/ x]^2 + b^3*(-c^4 + x^4)*ArcTan[c/x]^3 + b*ArcTan[c/x]*(2*a*b*c*x*(-3*c^2 + x^2) + b^2*c^2*(c^2 + x^2) + 3*a^2*(-c^4 + x^4) + 8*b^2*c^4*Log[1 - E^((2 *I)*ArcTan[c/x])]) + 8*a*b^2*c^4*Log[c/(Sqrt[1 + c^2/x^2]*x)] - (4*I)*b^3* c^4*PolyLog[2, E^((2*I)*ArcTan[c/x])])/4
Time = 1.76 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.41, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5363, 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.
| \(\displaystyle \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle -\int x^5 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3d\frac {1}{x}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \int \frac {x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (\int x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2d\frac {1}{x}-c^2 \int \frac {x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (-c^2 \int \frac {x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}+\frac {2}{3} b c \int \frac {x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (-c^2 \left (\int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2d\frac {1}{x}-c^2 \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {2}{3} b c \left (\int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )d\frac {1}{x}-c^2 \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \int \frac {x^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-c^2 \left (c^2 \left (-\int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+2 b c \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \left (c^2 \left (-\int \frac {1}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-x\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-c^2 \left (c^2 \left (-\int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+2 b c \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )-c^2 \left (c^2 \left (-\int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+2 b c \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (-c^2 \left (2 b c \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )+\frac {2}{3} b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (\frac {2}{3} b c \left (-\left (c^2 \left (i \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c}{x}+i}d\frac {1}{x}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )-c^2 \left (2 b c \left (i \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c}{x}+i}d\frac {1}{x}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (-c^2 \left (2 b c \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-\frac {i c}{x}}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )+\frac {2}{3} b c \left (-\left (c^2 \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-\frac {i c}{x}}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{4} b c \left (\frac {2}{3} b c \left (-\left (c^2 \left (i \left (-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (-c \arctan \left (\frac {c}{x}\right )-x\right )\right )-c^2 \left (2 b c \left (i \left (-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
Input:
Int[x^3*(a + b*ArcTan[c/x])^3,x]
Output:
(x^4*(a + b*ArcTan[c/x])^3)/4 - (3*b*c*(-1/3*(x^3*(a + b*ArcTan[c/x])^2) - c^2*(-(x*(a + b*ArcTan[c/x])^2) - (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*(x^2*(a + b*ArcTan[c/x])) + (b*c*(-x - 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
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_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTan[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplif y[(m + 1)/n]]
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. 479 vs. \(2 (196 ) = 392\).
Time = 7.85 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.24
| method | result | size |
| derivativedivides | \(-c^{4} \left (-\frac {a^{3} x^{4}}{4 c^{4}}+b^{3} \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )^{3}}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )^{3}}{4}-\frac {x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{4 c^{3}}+\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{4 c}+\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{4 c^{2}}-2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {\arctan \left (\frac {c}{x}\right )}{4}-\frac {x}{4 c}+\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{2}-i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )+i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )-i \operatorname {dilog}\left (1+\frac {i c}{x}\right )+i \operatorname {dilog}\left (1-\frac {i c}{x}\right )\right )+3 a \,b^{2} \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )^{2}}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )^{2}}{4}-\frac {x^{3} \arctan \left (\frac {c}{x}\right )}{6 c^{3}}+\frac {x \arctan \left (\frac {c}{x}\right )}{2 c}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {x^{2}}{12 c^{2}}-\frac {2 \ln \left (\frac {c}{x}\right )}{3}\right )+3 a^{2} b \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )}{4}-\frac {x^{3}}{12 c^{3}}+\frac {x}{4 c}\right )\right )\) | \(480\) |
| default | \(-c^{4} \left (-\frac {a^{3} x^{4}}{4 c^{4}}+b^{3} \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )^{3}}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )^{3}}{4}-\frac {x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{4 c^{3}}+\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{4 c}+\arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{4 c^{2}}-2 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {\arctan \left (\frac {c}{x}\right )}{4}-\frac {x}{4 c}+\frac {i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{2}-i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )+i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )-i \operatorname {dilog}\left (1+\frac {i c}{x}\right )+i \operatorname {dilog}\left (1-\frac {i c}{x}\right )\right )+3 a \,b^{2} \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )^{2}}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )^{2}}{4}-\frac {x^{3} \arctan \left (\frac {c}{x}\right )}{6 c^{3}}+\frac {x \arctan \left (\frac {c}{x}\right )}{2 c}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {x^{2}}{12 c^{2}}-\frac {2 \ln \left (\frac {c}{x}\right )}{3}\right )+3 a^{2} b \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )}{4}-\frac {x^{3}}{12 c^{3}}+\frac {x}{4 c}\right )\right )\) | \(480\) |
| parts | \(\frac {a^{3} x^{4}}{4}+\frac {b^{3} x^{4} \arctan \left (\frac {c}{x}\right )^{3}}{4}-\frac {b^{3} c^{4} \arctan \left (\frac {c}{x}\right )^{3}}{4}+\frac {b^{3} c^{3} x}{4}+\frac {3 a^{2} b \,x^{4} \arctan \left (\frac {c}{x}\right )}{4}-3 a \,b^{2} c^{4} \left (-\frac {x^{4} \arctan \left (\frac {c}{x}\right )^{2}}{4 c^{4}}+\frac {\arctan \left (\frac {c}{x}\right )^{2}}{4}-\frac {x^{3} \arctan \left (\frac {c}{x}\right )}{6 c^{3}}+\frac {x \arctan \left (\frac {c}{x}\right )}{2 c}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {x^{2}}{12 c^{2}}-\frac {2 \ln \left (\frac {c}{x}\right )}{3}\right )+\frac {a^{2} b c \,x^{3}}{4}-i b^{3} c^{4} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )-\frac {i b^{3} c^{4} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {i b^{3} c^{4} \ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {i b^{3} c^{4} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{2}+i b^{3} c^{4} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )-\frac {i b^{3} c^{4} \ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}-\frac {i b^{3} c^{4} \ln \left (\frac {c}{x}+i\right )^{2}}{4}+\frac {i b^{3} c^{4} \ln \left (\frac {c}{x}-i\right )^{2}}{4}-i b^{3} c^{4} \operatorname {dilog}\left (1-\frac {i c}{x}\right )-\frac {i b^{3} c^{4} \operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}+i b^{3} c^{4} \operatorname {dilog}\left (1+\frac {i c}{x}\right )+\frac {i b^{3} c^{4} \operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{2}-\frac {3 a^{2} b x \,c^{3}}{4}+\frac {3 a^{2} b \,c^{4} \arctan \left (\frac {x}{c}\right )}{4}-\frac {b^{3} c^{4} \arctan \left (\frac {x}{c}\right )}{4}+\frac {b^{3} c \,x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{4}-\frac {3 b^{3} c^{3} x \arctan \left (\frac {c}{x}\right )^{2}}{4}-b^{3} c^{4} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+\frac {b^{3} c^{2} x^{2} \arctan \left (\frac {c}{x}\right )}{4}+2 b^{3} c^{4} \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )\) | \(586\) |
| risch | \(\text {Expression too large to display}\) | \(246279\) |
Input:
int(x^3*(a+b*arctan(c/x))^3,x,method=_RETURNVERBOSE)
Output:
-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/c*x*arctan(c/x)^2+arctan(c/x)*ln(1+c^2/x^2)-1 /4/c^2*x^2*arctan(c/x)-2*ln(c/x)*arctan(c/x)-1/4*arctan(c/x)-1/4*x/c+1/2*I *(ln(c/x-I)*ln(1+c^2/x^2)-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(1+c^2/x^2)-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)))-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))+3*a*b^2*(-1/4/c^4*x^4*arct an(c/x)^2+1/4*arctan(c/x)^2-1/6/c^3*x^3*arctan(c/x)+1/2/c*x*arctan(c/x)+1/ 3*ln(1+c^2/x^2)-1/12/c^2*x^2-2/3*ln(c/x))+3*a^2*b*(-1/4/c^4*x^4*arctan(c/x )+1/4*arctan(c/x)-1/12/c^3*x^3+1/4*x/c))
\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*arctan(c/x))^3,x, algorithm="fricas")
Output:
integral(b^3*x^3*arctan(c/x)^3 + 3*a*b^2*x^3*arctan(c/x)^2 + 3*a^2*b*x^3*a rctan(c/x) + a^3*x^3, x)
\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x^{3} \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \] Input:
integrate(x**3*(a+b*atan(c/x))**3,x)
Output:
Integral(x**3*(a + b*atan(c/x))**3, x)
\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*arctan(c/x))^3,x, algorithm="maxima")
Output:
3/4*a*b^2*x^4*arctan(c/x)^2 + 1/4*a^3*x^4 + 1/4*(3*x^4*arctan(c/x) + (3*c^ 3*arctan(x/c) - 3*c^2*x + x^3)*c)*a^2*b + 1/4*((3*c^2*arctan(x/c)^2 - 4*c^ 2*log(c^2 + x^2) + x^2)*c^2 + 2*(3*c^3*arctan(x/c) - 3*c^2*x + x^3)*c*arct an(c/x))*a*b^2 - 1/64*(12*c^4*arctan(c/x)^2*arctan(x/c) + 8*c^4*arctan2(c, x)^3 - 8*x^4*arctan2(c, x)^3 + 4*(3*arctan(c/x)*arctan(x/c)^2/c + arctan( x/c)^3/c)*c^5 + 12*c^3*x*arctan2(c, x)^2 - 4*c*x^3*arctan2(c, x)^2 + 192*c ^5*integrate(1/64*log(c^2 + x^2)^2/(c^2 + x^2), x) + 1536*c^4*integrate(1/ 64*x*arctan(c/x)/(c^2 + x^2), x) + 768*c^3*integrate(1/64*x^2*log(c^2 + x^ 2)/(c^2 + x^2), x) - 2048*c^2*integrate(1/64*x^3*arctan(c/x)^3/(c^2 + x^2) , x) - 512*c^2*integrate(1/64*x^3*arctan(c/x)/(c^2 + x^2), x) - (3*c^3*x - c*x^3)*log(c^2 + x^2)^2 - 768*c*integrate(1/64*x^4*arctan(c/x)^2/(c^2 + x ^2), x) - 192*c*integrate(1/64*x^4*log(c^2 + x^2)^2/(c^2 + x^2), x) - 256* c*integrate(1/64*x^4*log(c^2 + x^2)/(c^2 + x^2), x) - 2048*integrate(1/64* x^5*arctan(c/x)^3/(c^2 + x^2), x))*b^3
\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*arctan(c/x))^3,x, algorithm="giac")
Output:
integrate((b*arctan(c/x) + a)^3*x^3, x)
Timed out. \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x^3\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \] Input:
int(x^3*(a + b*atan(c/x))^3,x)
Output:
int(x^3*(a + b*atan(c/x))^3, x)
\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=-\frac {\mathit {atan} \left (\frac {c}{x}\right )^{3} b^{3} c^{4}}{4}+\frac {\mathit {atan} \left (\frac {c}{x}\right )^{3} b^{3} x^{4}}{4}-\frac {3 \mathit {atan} \left (\frac {c}{x}\right )^{2} a \,b^{2} c^{4}}{4}+\frac {3 \mathit {atan} \left (\frac {c}{x}\right )^{2} a \,b^{2} x^{4}}{4}-\frac {3 \mathit {atan} \left (\frac {c}{x}\right )^{2} b^{3} c^{3} x}{4}+\frac {\mathit {atan} \left (\frac {c}{x}\right )^{2} b^{3} c \,x^{3}}{4}-\frac {3 \mathit {atan} \left (\frac {c}{x}\right ) a^{2} b \,c^{4}}{4}+\frac {3 \mathit {atan} \left (\frac {c}{x}\right ) a^{2} b \,x^{4}}{4}-\frac {3 \mathit {atan} \left (\frac {c}{x}\right ) a \,b^{2} c^{3} x}{2}+\frac {\mathit {atan} \left (\frac {c}{x}\right ) a \,b^{2} c \,x^{3}}{2}+\frac {\mathit {atan} \left (\frac {c}{x}\right ) b^{3} c^{4}}{4}+\frac {\mathit {atan} \left (\frac {c}{x}\right ) b^{3} c^{2} x^{2}}{4}-2 \left (\int \frac {\mathit {atan} \left (\frac {c}{x}\right ) x}{c^{2}+x^{2}}d x \right ) b^{3} c^{4}-\mathrm {log}\left (c^{2}+x^{2}\right ) a \,b^{2} c^{4}+\frac {a^{3} x^{4}}{4}-\frac {3 a^{2} b \,c^{3} x}{4}+\frac {a^{2} b c \,x^{3}}{4}+\frac {a \,b^{2} c^{2} x^{2}}{4}+\frac {b^{3} c^{3} x}{4} \] Input:
int(x^3*(a+b*atan(c/x))^3,x)
Output:
( - atan(c/x)**3*b**3*c**4 + atan(c/x)**3*b**3*x**4 - 3*atan(c/x)**2*a*b** 2*c**4 + 3*atan(c/x)**2*a*b**2*x**4 - 3*atan(c/x)**2*b**3*c**3*x + atan(c/ x)**2*b**3*c*x**3 - 3*atan(c/x)*a**2*b*c**4 + 3*atan(c/x)*a**2*b*x**4 - 6* atan(c/x)*a*b**2*c**3*x + 2*atan(c/x)*a*b**2*c*x**3 + atan(c/x)*b**3*c**4 + atan(c/x)*b**3*c**2*x**2 - 8*int((atan(c/x)*x)/(c**2 + x**2),x)*b**3*c** 4 - 4*log(c**2 + x**2)*a*b**2*c**4 + a**3*x**4 - 3*a**2*b*c**3*x + a**2*b* c*x**3 + a*b**2*c**2*x**2 + b**3*c**3*x)/4