\(\int (e+f x)^2 (a+b \cot ^{-1}(c+d x))^3 \, dx\) [141]

Optimal result

Integrand size = 20, antiderivative size = 565 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \]

[Out]

a*b^2*f^2*x/d^2+b^3*f^2*(d*x+c)*arccot(d*x+c)/d^3+1/2*b*f^2*(a+b*arccot(d*x+c))^2/d^3+3*I*b*f*(-c*f+d*e)*(a+b*
arccot(d*x+c))^2/d^3+3*b*f*(-c*f+d*e)*(d*x+c)*(a+b*arccot(d*x+c))^2/d^3+1/2*b*f^2*(d*x+c)^2*(a+b*arccot(d*x+c)
)^2/d^3+1/3*I*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arccot(d*x+c))^3/d^3-1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f
-(-c^2+3)*f^2)*(a+b*arccot(d*x+c))^3/d^3/f+1/3*(f*x+e)^3*(a+b*arccot(d*x+c))^3/f-6*b^2*f*(-c*f+d*e)*(a+b*arcco
t(d*x+c))*ln(2/(1+I*(d*x+c)))/d^3-b*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arccot(d*x+c))^2*ln(2/(1+I*(d*x+
c)))/d^3+1/2*b^3*f^2*ln(1+(d*x+c)^2)/d^3+3*I*b^3*f*(-c*f+d*e)*polylog(2,1-2/(1+I*(d*x+c)))/d^3+I*b^2*(3*d^2*e^
2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arccot(d*x+c))*polylog(2,1-2/(1+I*(d*x+c)))/d^3-1/2*b^3*(3*d^2*e^2-6*c*d*e*f-
(-3*c^2+1)*f^2)*polylog(3,1-2/(1+I*(d*x+c)))/d^3

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5156, 4975, 4931, 5041, 4965, 2449, 2352, 4947, 5037, 266, 5005, 5105, 5115, 6745} \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}+\frac {a b^2 f^2 x}{d^2}+\frac {i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}-\frac {b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (c+d x) (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{d^3}+\frac {b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3} \]

[In]

Int[(e + f*x)^2*(a + b*ArcCot[c + d*x])^3,x]

[Out]

(a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + ((3*
I)*b*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/d^
3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + ((I/3)*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(
a + b*ArcCot[c + d*x])^3)/d^3 - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcCot[c + d*x])^3)/
(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])*Log[2
/(1 + I*(c + d*x))])/d^3 - (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I
*(c + d*x))])/d^3 + (b^3*f^2*Log[1 + (c + d*x)^2])/(2*d^3) + ((3*I)*b^3*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*
(c + d*x))])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(
1 + I*(c + d*x))])/d^3 - (b^3*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(
2*d^3)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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]

Rule 4931

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Dist[b*c
*n*p, Int[x^n*((a + b*ArcCot[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])

Rule 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[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 4965

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

Rule 4975

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a
+ b*ArcCot[c*x])^p/(e*(q + 1))), x] + Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcCot[c*x])^(p -
1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && N
eQ[q, -1]

Rule 5005

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(a + b*ArcCot[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 5037

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcCot[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5041

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*((a + b*ArcCot[
c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c
, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5105

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcCot[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rule 5115

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*p*(I/2), Int[(a + b*ArcCot[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2
*(I/(I - c*x)))^2, 0]

Rule 5156

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \cot ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \text {Subst}\left (\int \left (\frac {3 f^2 (d e-c f) \left (a+b \cot ^{-1}(x)\right )^2}{d^3}+\frac {f^3 x \left (a+b \cot ^{-1}(x)\right )^2}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) \left (a+b \cot ^{-1}(x)\right )^2}{d^3 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f} \\ & = \frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \text {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b f^2\right ) \text {Subst}\left (\int x \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}+\frac {(3 b f (d e-c f)) \text {Subst}\left (\int \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3} \\ & = \frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \text {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2}+\frac {f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {x \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^3}+\frac {\left (b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (b (d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d^3 f} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \cot ^{-1}(x) \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{i-x} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 i b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^3}-\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (i b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2309\) vs. \(2(565)=1130\).

Time = 14.56 (sec) , antiderivative size = 2309, normalized size of antiderivative = 4.09 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\text {Result too large to show} \]

[In]

Integrate[(e + f*x)^2*(a + b*ArcCot[c + d*x])^3,x]

[Out]

(a^2*(a*d^2*e^2 + 3*b*d*e*f - 2*b*c*f^2)*x)/d^2 + (a^2*f*(2*a*d*e + b*f)*x^2)/(2*d) + (a^3*f^2*x^3)/3 + a^2*b*
x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[c + d*x] + ((-3*a^2*b*c*d^2*e^2 - 3*a^2*b*d*e*f + 3*a^2*b*c^2*d*e*f + 3*a
^2*b*c*f^2 - a^2*b*c^3*f^2)*ArcTan[c + d*x])/d^3 + ((3*a^2*b*d^2*e^2 - 6*a^2*b*c*d*e*f - a^2*b*f^2 + 3*a^2*b*c
^2*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d^3) - (3*a*b^2*e^2*(1 + (c + d*x)^2)*(-((c + d*x)*ArcCot[c + d*x
]^2) + 2*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - I*(ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCo
t[c + d*x])])))/(d*(c + d*x)^2*(1 + (c + d*x)^(-2))) + (6*a*b^2*e*f*(1 + (c + d*x)^2)*(((c + d*x)*ArcCot[c + d
*x])/d^2 - (c*(c + d*x)*ArcCot[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 + (c + d*x)^(-2))*ArcCot[c + d*x]^2)/(2*d^2)
- Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]/d^2 + (2*c*(ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] -
(I/2)*(ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/d^2))/((c + d*x)^2*(1 + (c + d*x)^(-2)))
- (a*b^2*f^2*x^2*(1 + (c + d*x)^2)*(-((c + d*x)*ArcCot[c + d*x]^2) + ArcCot[c + d*x]*(-1 + 3*c*ArcCot[c + d*x]
) - (1 - 6*c*ArcCot[c + d*x] - ArcCot[c + d*x]^2 + 3*c^2*ArcCot[c + d*x]^2)/((c + d*x)*(1 + (c + d*x)^(-2))) -
(6*c*(Log[(c + d*x)^(-1)] + Log[1/Sqrt[1 + (c + d*x)^(-2)]]))/((c + d*x)^2*(1 + (c + d*x)^(-2))) + (I*(ArcCot
[c + d*x]*(ArcCot[c + d*x] + (2*I)*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + PolyLog[2, E^((2*I)*ArcCot[c + d*x])]
))/((c + d*x)^2*(1 + (c + d*x)^(-2))) + (6*c^2*(ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - (I/2)*(Ar
cCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/((c + d*x)^2*(1 + (c + d*x)^(-2)))))/(d*(c + d*x)^2
*(1 + (c + d*x)^(-2))*(1/Sqrt[1 + (c + d*x)^(-2)] - c/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))^2) - (b^3*e^2*(1 +
(c + d*x)^2)*((-1/8*I)*Pi^3 + I*ArcCot[c + d*x]^3 - (c + d*x)*ArcCot[c + d*x]^3 + 3*ArcCot[c + d*x]^2*Log[1 -
E^((-2*I)*ArcCot[c + d*x])] + (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] + (3*PolyLog[3, E^
((-2*I)*ArcCot[c + d*x])])/2))/(d*(c + d*x)^2*(1 + (c + d*x)^(-2))) + (b^3*e*f*(1 + (c + d*x)^2)*((-I)*c*Pi^3
+ (12*I)*ArcCot[c + d*x]^2 + 12*(c + d*x)*ArcCot[c + d*x]^2 + (8*I)*c*ArcCot[c + d*x]^3 - 8*c*(c + d*x)*ArcCot
[c + d*x]^3 + 4*(c + d*x)^2*(1 + (c + d*x)^(-2))*ArcCot[c + d*x]^3 + 24*c*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*
ArcCot[c + d*x])] - 24*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (24*I)*c*ArcCot[c + d*x]*PolyLog[2
, E^((-2*I)*ArcCot[c + d*x])] + (12*I)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])] + 12*c*PolyLog[3, E^((-2*I)*ArcCo
t[c + d*x])]))/(4*d^2*(c + d*x)^2*(1 + (c + d*x)^(-2))) - (b^3*f^2*(1 + (c + d*x)^2)*(I*(-1 + 3*c^2)*ArcCot[c
+ d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] + ((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)*(((3*I)*Pi^3)/((c + d*
x)*Sqrt[1 + (c + d*x)^(-2)]) - ((9*I)*c^2*Pi^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (24*ArcCot[c + d*x])/Sq
rt[1 + (c + d*x)^(-2)] + (72*c*ArcCot[c + d*x]^2)/Sqrt[1 + (c + d*x)^(-2)] - (48*ArcCot[c + d*x]^2)/((c + d*x)
*Sqrt[1 + (c + d*x)^(-2)]) + ((216*I)*c*ArcCot[c + d*x]^2)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (24*ArcCot[c
+ d*x]^3)/Sqrt[1 + (c + d*x)^(-2)] - (24*c^2*ArcCot[c + d*x]^3)/Sqrt[1 + (c + d*x)^(-2)] - ((24*I)*ArcCot[c +
d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (96*c*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])
+ ((72*I)*c^2*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + 24*ArcCot[c + d*x]*Cos[3*ArcCot[c + d*
x]] - 72*c*ArcCot[c + d*x]^2*Cos[3*ArcCot[c + d*x]] - 8*ArcCot[c + d*x]^3*Cos[3*ArcCot[c + d*x]] + 24*c^2*ArcC
ot[c + d*x]^3*Cos[3*ArcCot[c + d*x]] - (72*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)*S
qrt[1 + (c + d*x)^(-2)]) + (216*c^2*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 +
(c + d*x)^(-2)]) - (432*c*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c + d*x)^(
-2)]) + (72*Log[(c + d*x)^(-1)])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (72*Log[1/Sqrt[1 + (c + d*x)^(-2)]])/(
(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + ((288*I)*c*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/((c + d*x)^3*(1 + (c +
d*x)^(-2))^(3/2)) + (48*(-1 + 3*c^2)*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)^3*(1 + (c + d*x)^(-2)
)^(3/2)) - I*Pi^3*Sin[3*ArcCot[c + d*x]] + (3*I)*c^2*Pi^3*Sin[3*ArcCot[c + d*x]] - (72*I)*c*ArcCot[c + d*x]^2*
Sin[3*ArcCot[c + d*x]] + (8*I)*ArcCot[c + d*x]^3*Sin[3*ArcCot[c + d*x]] - (24*I)*c^2*ArcCot[c + d*x]^3*Sin[3*A
rcCot[c + d*x]] + 24*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] - 72*c^2*Arc
Cot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] + 144*c*ArcCot[c + d*x]*Log[1 - E^((
2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] - 24*Log[(c + d*x)^(-1)]*Sin[3*ArcCot[c + d*x]] - 24*Log[1/Sqrt[
1 + (c + d*x)^(-2)]]*Sin[3*ArcCot[c + d*x]]))/96))/(d^3*(c + d*x)^2*(1 + (c + d*x)^(-2)))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 101.64 (sec) , antiderivative size = 6248, normalized size of antiderivative = 11.06

method	result	size
parts	\(\text {Expression too large to display}\)	\(6248\)
derivativedivides	\(\text {Expression too large to display}\)	\(10834\)
default	\(\text {Expression too large to display}\)	\(10834\)

[In]

int((f*x+e)^2*(a+b*arccot(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((f*x+e)^2*(a+b*arccot(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(a^3*f^2*x^2 + 2*a^3*e*f*x + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*e*f*x + b^3*e^2)*arccot(d*x + c)^3 + 3*(a*
b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^2)*arccot(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*arc
cot(d*x + c), x)

Sympy [F(-1)]

Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**2*(a+b*acot(d*x+c))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((f*x+e)^2*(a+b*arccot(d*x+c))^3,x, algorithm="maxima")

[Out]

1/24*b^3*f^2*x^3*arctan2(1, d*x + c)^3 + 1/8*b^3*e*f*x^2*arctan2(1, d*x + c)^3 + 1/8*b^3*e^2*x*arctan2(1, d*x
+ c)^3 + 1/3*a^3*f^2*x^3 + a^3*e*f*x^2 + 3*(x^2*arccot(d*x + c) + d*(x/d^2 + (c^2 - 1)*arctan((d^2*x + c*d)/d)
/d^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^3))*a^2*b*e*f + 1/2*(2*x^3*arccot(d*x + c) + d*((d*x^2 - 4*c*x)/d^
3 - 2*(c^3 - 3*c)*arctan((d^2*x + c*d)/d)/d^4 + (3*c^2 - 1)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^4))*a^2*b*f^2 +
a^3*e^2*x + 3/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a^2*b*e^2/d - 1/32*(b^3*f^2*x^3*arctan2(
1, d*x + c) + 3*b^3*e*f*x^2*arctan2(1, d*x + c) + 3*b^3*e^2*x*arctan2(1, d*x + c))*log(d^2*x^2 + 2*c*d*x + c^2
+ 1)^2 + integrate(1/32*(4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*f^2*x^4 + 4*(2*
(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*e*f + (b^3*arctan2(1, d*x + c)^2 + 2*(7*b^3
*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*f^2)*x^3 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a
*b^2*arctan2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*e^2 + 4*((7*b
^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*e^2 + (3*b^3*arctan2(1, d*x + c)^2 + 4*(7*b^3*a
rctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*e*f + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arct
an2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*f^2)*x^2 + (3*b^3*d^2*
f^2*x^4*arctan2(1, d*x + c) + (6*b^3*d^2*e*f*arctan2(1, d*x + c) + (6*b^3*c*arctan2(1, d*x + c) - b^3)*d*f^2)*
x^3 + 3*(b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*e^2 + 3*(b^3*d^2*e^2*arctan2(1, d*x + c) + (4*
b^3*c*arctan2(1, d*x + c) - b^3)*d*e*f + (b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*f^2)*x^2 + 3*
((2*b^3*c*arctan2(1, d*x + c) - b^3)*d*e^2 + 2*(b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*e*f)*x)
*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*((3*b^3*arctan2(1, d*x + c)^2 + 2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*
b^2*arctan2(1, d*x + c)^2)*c)*d*e^2 + 2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2 + (7*b^3
*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*e*f)*x + 4*(b^3*d^2*f^2*x^4*arctan2(1, d*x + c)
+ 3*b^3*c*d*e^2*x*arctan2(1, d*x + c) + (3*b^3*d^2*e*f*arctan2(1, d*x + c) + b^3*c*d*f^2*arctan2(1, d*x + c))*
x^3 + 3*(b^3*d^2*e^2*arctan2(1, d*x + c) + b^3*c*d*e*f*arctan2(1, d*x + c))*x^2)*log(d^2*x^2 + 2*c*d*x + c^2 +
1))/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)

Giac [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((f*x+e)^2*(a+b*arccot(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^2*(b*arccot(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((e + f*x)^2*(a + b*acot(c + d*x))^3,x)

[Out]

int((e + f*x)^2*(a + b*acot(c + d*x))^3, x)