\(\int (e+f x) (a+b \cot ^{-1}(c+d x))^3 \, dx\) [142]

Optimal result

Integrand size = 18, antiderivative size = 337 \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5156, 4975, 4931, 5041, 4965, 2449, 2352, 5105, 5005, 5115, 6745} \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {3 i b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}-\frac {3 b^2 f \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}-\frac {3 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}+\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d^2}+\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{2 d^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 5.42 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.87 \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {a^2 (2 a d e+3 b f-2 a c f) (c+d x)+a^3 f (c+d x)^2-3 a^2 b (c+d x) (c f-d (2 e+f x)) \cot ^{-1}(c+d x)-3 a^2 b f \arctan (c+d x)+6 a b^2 f \left ((c+d x) \cot ^{-1}(c+d x)+\frac {1}{2} \left (1+(c+d x)^2\right ) \cot ^{-1}(c+d x)^2-\log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )\right )+3 a^2 b (d e-c f) \log \left (1+(c+d x)^2\right )+6 a b^2 d e \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )-6 a b^2 c f \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )+b^3 f \left (3 (c+d x) \cot ^{-1}(c+d x)^2+\left (1+(c+d x)^2\right ) \cot ^{-1}(c+d x)^3-6 \cot ^{-1}(c+d x) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+3 i \left (\cot ^{-1}(c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )+2 b^3 d e \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )-3 i \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )\right )-2 b^3 c f \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )-3 i \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )\right )}{2 d^2} \]

[In]

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

[Out]

(a^2*(2*a*d*e + 3*b*f - 2*a*c*f)*(c + d*x) + a^3*f*(c + d*x)^2 - 3*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcCo
t[c + d*x] - 3*a^2*b*f*ArcTan[c + d*x] + 6*a*b^2*f*((c + d*x)*ArcCot[c + d*x] + ((1 + (c + d*x)^2)*ArcCot[c +
d*x]^2)/2 - Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]) + 3*a^2*b*(d*e - c*f)*Log[1 + (c + d*x)^2] + 6*a*b^2*
d*e*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^(
(2*I)*ArcCot[c + d*x])]) - 6*a*b^2*c*f*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*Ar
cCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]) + b^3*f*(3*(c + d*x)*ArcCot[c + d*x]^2 + (1 + (c +
d*x)^2)*ArcCot[c + d*x]^3 - 6*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (3*I)*(ArcCot[c + d*x]^2 +
PolyLog[2, E^((2*I)*ArcCot[c + d*x])])) + 2*b^3*d*e*((I/8)*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) - 2*b^3*c*f*((I/8)*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))/(2*d^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (316 ) = 632\).

Time = 13.32 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.12

method	result	size
parts	\(\text {Expression too large to display}\)	\(1051\)
derivativedivides	\(\text {Expression too large to display}\)	\(17316\)
default	\(\text {Expression too large to display}\)	\(17316\)

[In]

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

[Out]

a^3*(1/2*f*x^2+e*x)+b^3/d*(1/2/d*arccot(d*x+c)^3*(d*x+c)^2*f-1/d*arccot(d*x+c)^3*c*f*(d*x+c)+arccot(d*x+c)^3*e
*(d*x+c)+3/2/d*(1/3*f*arccot(d*x+c)^3+2*I*f*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-2/3*I*arccot(d*x+c)^3*c*
f+2*I*f*arccot(d*x+c)^2+arccot(d*x+c)^2*f*(d*x+c-I)-2*f*arccot(d*x+c)*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+2/3*
I*arccot(d*x+c)^3*d*e-2*f*arccot(d*x+c)*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+4*polylog(3,-(d*x+c+I)/(1+(d*x+c)^
2)^(1/2))*c*f+4*I*d*e*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1
/2))*c*f*arccot(d*x+c)^2+2*I*f*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*
d*e*arccot(d*x+c)^2-4*I*c*f*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-4*polylog(3,-(d*x+c+I)/(1+(
d*x+c)^2)^(1/2))*d*e+4*I*d*e*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-4*I*c*f*arccot(d*x+c)*pol
ylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-4*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*d*e+4*polylog(3,(d*x+c+I)/(1
+(d*x+c)^2)^(1/2))*c*f+2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*c*f*arccot(d*x+c)^2-2*ln(1+(d*x+c+I)/(1+(d*x+c)^2
)^(1/2))*d*e*arccot(d*x+c)^2))+3*a*b^2/d*(1/2/d*arccot(d*x+c)^2*(d*x+c)^2*f-1/d*arccot(d*x+c)^2*c*f*(d*x+c)+ar
ccot(d*x+c)^2*e*(d*x+c)+1/d*(-arccot(d*x+c)*ln(1+(d*x+c)^2)*c*f+arccot(d*x+c)*ln(1+(d*x+c)^2)*d*e-arccot(d*x+c
)*arctan(d*x+c)*f+arccot(d*x+c)*(d*x+c)*f+1/2*f*ln(1+(d*x+c)^2)-1/2*arctan(d*x+c)^2*f+1/2*(-2*c*f+2*d*e)*(-1/2
*I*(ln(d*x+c-I)*ln(1+(d*x+c)^2)-1/2*ln(d*x+c-I)^2-dilog(-1/2*I*(d*x+c+I))-ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I)))+1/
2*I*(ln(d*x+c+I)*ln(1+(d*x+c)^2)-1/2*ln(d*x+c+I)^2-dilog(1/2*I*(d*x+c-I))-ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))))))+
3*a^2*b/d*(1/2/d*arccot(d*x+c)*(d*x+c)^2*f-1/d*arccot(d*x+c)*c*f*(d*x+c)+arccot(d*x+c)*e*(d*x+c)+1/2/d*(f*(d*x
+c)+1/2*(-2*c*f+2*d*e)*ln(1+(d*x+c)^2)-f*arctan(d*x+c)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/16*b^3*f*x^2*arctan2(1, d*x + c)^3 + 1/8*b^3*e*x*arctan2(1, d*x + c)^3 + 1/2*a^3*f*x^2 + 3/2*(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*f +
a^3*e*x + 3/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a^2*b*e/d - 3/64*(b^3*f*x^2*arctan2(1, d*x
+ c) + 2*b^3*e*x*arctan2(1, d*x + c))*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + integrate(1/64*(8*(7*b^3*arctan2(1,
d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*f*x^3 + 4*(2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2
(1, d*x + c)^2)*d^2*e + (3*b^3*arctan2(1, d*x + c)^2 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*
x + c)^2)*c)*d*f)*x^2 + 3*(2*b^3*d^2*f*x^3*arctan2(1, d*x + c) + (2*b^3*d^2*e*arctan2(1, d*x + c) + (4*b^3*c*a
rctan2(1, d*x + c) - b^3)*d*f)*x^2 + 2*(b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*e + 2*((2*b^3*c
*arctan2(1, d*x + c) - b^3)*d*e + (b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*f)*x)*log(d^2*x^2 +
2*c*d*x + c^2 + 1)^2 + 8*(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 + 8*((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 + (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)*f)*x + 12*(b^3*d^2*f*x^3*arctan2(1, d*x
+ c) + 2*b^3*c*d*e*x*arctan2(1, d*x + c) + (2*b^3*d^2*e*arctan2(1, d*x + c) + b^3*c*d*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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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