\(\int x^3 (a+b \arctan (\frac {c}{x}))^3 \, dx\) [147]

Optimal result

Integrand size = 16, antiderivative size = 214 \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{4} b^3 c^3 x+\frac {1}{4} b^3 c^4 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-i b c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {3}{4} b c^3 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} b c x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+2 b^2 c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )-i b^3 c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \]

[Out]

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))

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4948, 4946, 5038, 331, 209, 5044, 4988, 2497, 5004} \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=2 b^2 c^4 \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-i b c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {3}{4} b c^3 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} b c x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-i b^3 c^4 \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right )+\frac {1}{4} b^3 c^4 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^3 c^3 x \]

[In]

Int[x^3*(a + b*ArcTan[c/x])^3,x]

[Out]

(b^3*c^3*x)/4 + (b^3*c^4*ArcCot[x/c])/4 + (b^2*c^2*x^2*(a + b*ArcCot[x/c]))/4 - I*b*c^4*(a + b*ArcCot[x/c])^2
- (3*b*c^3*x*(a + b*ArcCot[x/c])^2)/4 + (b*c*x^3*(a + b*ArcCot[x/c])^2)/4 - (c^4*(a + b*ArcCot[x/c])^3)/4 + (x
^4*(a + b*ArcCot[x/c])^3)/4 + 2*b^2*c^4*(a + b*ArcCot[x/c])*Log[2 - 2/(1 - (I*c)/x)] - I*b^3*c^4*PolyLog[2, -1
+ 2/(1 - (I*c)/x)]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2497

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[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4946

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] - Dist[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]

Rule 4948

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[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[Sim
plify[(m + 1)/n]]

Rule 4988

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] - Dist[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]

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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]

Rule 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[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]

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b \arctan (c x))^3}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\frac {1}{4} (3 b c) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^4 \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\frac {1}{4} (3 b c) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^4} \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 b c^3\right ) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^2 \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{4} b c x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\frac {1}{2} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x^3 \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 b c^3\right ) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (3 b c^5\right ) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\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-\frac {1}{2} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x^3} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 b^2 c^4\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\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-\frac {1}{4} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (i b^2 c^4\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 i b^2 c^4\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{4} b^3 c^3 x+\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 )+\frac {1}{4} \left (b^3 c^5\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \left (b^3 c^5\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \left (3 b^3 c^5\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.18 \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{4} \left (a b^2 c^4-3 a^2 b c^3 x+b^3 c^3 x+a b^2 c^2 x^2+a^2 b c x^3+a^3 x^4+b^2 \left (b c \left (-4 i c^3-3 c^2 x+x^3\right )+3 a \left (-c^4+x^4\right )\right ) \arctan \left (\frac {c}{x}\right )^2+b^3 \left (-c^4+x^4\right ) \arctan \left (\frac {c}{x}\right )^3+b \arctan \left (\frac {c}{x}\right ) \left (2 a b c x \left (-3 c^2+x^2\right )+b^2 c^2 \left (c^2+x^2\right )+3 a^2 \left (-c^4+x^4\right )+8 b^2 c^4 \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )+8 a b^2 c^4 \log \left (\frac {c}{\sqrt {1+\frac {c^2}{x^2}} x}\right )-4 i b^3 c^4 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right ) \]

[In]

Integrate[x^3*(a + b*ArcTan[c/x])^3,x]

[Out]

(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*L
og[c/(Sqrt[1 + c^2/x^2]*x)] - (4*I)*b^3*c^4*PolyLog[2, E^((2*I)*ArcTan[c/x])])/4

Maple [B] (verified)

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 = 14.52 (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}-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 )+\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}\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 {x^{3}}{12 c^{3}}+\frac {x}{4 c}+\frac {\arctan \left (\frac {c}{x}\right )}{4}\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}-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 )+\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}\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 {x^{3}}{12 c^{3}}+\frac {x}{4 c}+\frac {\arctan \left (\frac {c}{x}\right )}{4}\right )\right )\)	\(480\)
parts	\(\frac {a^{2} b c \,x^{3}}{4}+\frac {a^{3} x^{4}}{4}+\frac {b^{3} c^{3} x}{4}-\frac {c^{4} b^{3} \arctan \left (\frac {x}{c}\right )}{4}-i c^{4} b^{3} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )-\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+i c^{4} b^{3} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )+\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}-\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}+\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{2}+\frac {3 b \,a^{2} c^{4} \arctan \left (\frac {x}{c}\right )}{4}-\frac {3 a^{2} b x \,c^{3}}{4}+\frac {b^{3} x^{4} \arctan \left (\frac {c}{x}\right )^{3}}{4}-\frac {c^{4} b^{3} \arctan \left (\frac {c}{x}\right )^{3}}{4}+\frac {c \,b^{3} \arctan \left (\frac {c}{x}\right )^{2} x^{3}}{4}-\frac {3 c^{3} b^{3} \arctan \left (\frac {c}{x}\right )^{2} x}{4}+\frac {c^{2} b^{3} x^{2} \arctan \left (\frac {c}{x}\right )}{4}+2 c^{4} b^{3} \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}+i\right )^{2}}{4}-i c^{4} b^{3} \operatorname {dilog}\left (1-\frac {i c}{x}\right )+\frac {i c^{4} b^{3} \operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{2}-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 )-c^{4} b^{3} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+\frac {3 a^{2} b \,x^{4} \arctan \left (\frac {c}{x}\right )}{4}-\frac {i c^{4} b^{3} \operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}+i c^{4} b^{3} \operatorname {dilog}\left (1+\frac {i c}{x}\right )+\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}-i\right )^{2}}{4}\)	\(586\)
risch	\(\text {Expression too large to display}\)	\(246279\)

[In]

int(x^3*(a+b*arctan(c/x))^3,x,method=_RETURNVERBOSE)

[Out]

-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*arc
tan(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-I*l
n(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(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))))+3*a*b^2*(-1/4/c^4*x^4*arctan(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*arct
an(c/x)-1/12/c^3*x^3+1/4*x/c+1/4*arctan(c/x)))

Fricas [F]

\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3} \,d x } \]

[In]

integrate(x^3*(a+b*arctan(c/x))^3,x, algorithm="fricas")

[Out]

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*arctan(c/x) + a^3*x^3, x)

Sympy [F]

\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x^{3} \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]

[In]

integrate(x**3*(a+b*atan(c/x))**3,x)

[Out]

Integral(x**3*(a + b*atan(c/x))**3, x)

Maxima [F]

\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3} \,d x } \]

[In]

integrate(x^3*(a+b*arctan(c/x))^3,x, algorithm="maxima")

[Out]

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*ar
ctan(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 + 1
92*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

Giac [F]

\[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3} \,d x } \]

[In]

integrate(x^3*(a+b*arctan(c/x))^3,x, algorithm="giac")

[Out]

integrate((b*arctan(c/x) + a)^3*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x^3\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \]

[In]

int(x^3*(a + b*atan(c/x))^3,x)

[Out]

int(x^3*(a + b*atan(c/x))^3, x)