\(\int (e+f x)^2 (a+b \cot ^{-1}(c+d x))^2 \, dx\) [136]

Optimal result

Integrand size = 20, antiderivative size = 382 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \arctan (c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5156, 4975, 4931, 266, 4947, 327, 209, 5105, 5005, 5041, 4965, 2449, 2352} \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}-\frac {2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {2 a b f x (d e-c f)}{d^2}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \arctan (c+d x)}{3 d^3}+\frac {i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}+\frac {2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}+\frac {b^2 f^2 x}{3 d^2} \]

[In]

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

[Out]

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

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 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 327

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

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 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 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]

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

Mathematica [A] (verified)

Time = 6.67 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.53 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 c f^2+3 a^2 d^3 e^2 x+6 a b d^2 e f x+b^2 d f^2 x-4 a b c d f^2 x+3 a^2 d^3 e f x^2+a b d^2 f^2 x^2+a^2 d^3 f^2 x^3+b^2 (i+c+d x) \left ((i+c)^2 f^2-(i+c) d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \cot ^{-1}(c+d x)^2-6 a b c d^2 e^2 \arctan (c+d x)-6 a b d e f \arctan (c+d x)+6 a b c^2 d e f \arctan (c+d x)+6 a b c f^2 \arctan (c+d x)-2 a b c^3 f^2 \arctan (c+d x)-b \cot ^{-1}(c+d x) \left (-2 a d^3 x \left (3 e^2+3 e f x+f^2 x^2\right )+b f \left (5 c^2 f-6 d^2 e x+c d (-6 e+4 f x)-f \left (1+d^2 x^2\right )\right )+2 b \left (3 d^2 e^2-6 c d e f+\left (-1+3 c^2\right ) f^2\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+6 b^2 c f^2 \log \left (\frac {1}{c+d x}\right )+3 a b d^2 e^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )-6 a b c d e f \log \left (1+c^2+2 c d x+d^2 x^2\right )-a b f^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )+3 a b c^2 f^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )+6 b^2 c f^2 \log \left (\frac {1}{\sqrt {1+\frac {1}{(c+d x)^2}}}\right )-6 b^2 d e f \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+i b^2 \left (3 d^2 e^2-6 c d e f+\left (-1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )}{3 d^3} \]

[In]

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

[Out]

(b^2*c*f^2 + 3*a^2*d^3*e^2*x + 6*a*b*d^2*e*f*x + b^2*d*f^2*x - 4*a*b*c*d*f^2*x + 3*a^2*d^3*e*f*x^2 + a*b*d^2*f
^2*x^2 + a^2*d^3*f^2*x^3 + b^2*(I + c + d*x)*((I + c)^2*f^2 - (I + c)*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x +
f^2*x^2))*ArcCot[c + d*x]^2 - 6*a*b*c*d^2*e^2*ArcTan[c + d*x] - 6*a*b*d*e*f*ArcTan[c + d*x] + 6*a*b*c^2*d*e*f
*ArcTan[c + d*x] + 6*a*b*c*f^2*ArcTan[c + d*x] - 2*a*b*c^3*f^2*ArcTan[c + d*x] - b*ArcCot[c + d*x]*(-2*a*d^3*x
*(3*e^2 + 3*e*f*x + f^2*x^2) + b*f*(5*c^2*f - 6*d^2*e*x + c*d*(-6*e + 4*f*x) - f*(1 + d^2*x^2)) + 2*b*(3*d^2*e
^2 - 6*c*d*e*f + (-1 + 3*c^2)*f^2)*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + 6*b^2*c*f^2*Log[(c + d*x)^(-1)] + 3*a
*b*d^2*e^2*Log[1 + c^2 + 2*c*d*x + d^2*x^2] - 6*a*b*c*d*e*f*Log[1 + c^2 + 2*c*d*x + d^2*x^2] - a*b*f^2*Log[1 +
c^2 + 2*c*d*x + d^2*x^2] + 3*a*b*c^2*f^2*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + 6*b^2*c*f^2*Log[1/Sqrt[1 + (c + d
*x)^(-2)]] - 6*b^2*d*e*f*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + I*b^2*(3*d^2*e^2 - 6*c*d*e*f + (-1 + 3*
c^2)*f^2)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/(3*d^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (362 ) = 724\).

Time = 1.34 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.81

method	result	size
parts	\(\text {Expression too large to display}\)	\(1072\)
derivativedivides	\(\text {Expression too large to display}\)	\(1087\)
default	\(\text {Expression too large to display}\)	\(1087\)
risch	\(\text {Expression too large to display}\)	\(3165\)

[In]

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

[Out]

1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arccot(d*x+c)^2*(d*x+c)^3-1/d^2*f^2*arccot(d*x+c)^2*(d*x+c)^2*c+1/d*f*a
rccot(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^2*arccot(d*x+c)^2*(d*x+c)*c^2-2/d*f*arccot(d*x+c)^2*(d*x+c)*c*e+arccot(d*x+
c)^2*(d*x+c)*e^2-1/3/d^2*f^2*arccot(d*x+c)^2*c^3+1/d*f*arccot(d*x+c)^2*c^2*e-arccot(d*x+c)^2*c*e^2+1/3*d/f*arc
cot(d*x+c)^2*e^3+2/3/d^2/f*(1/2*arccot(d*x+c)*f^3*(d*x+c)^2-3*arccot(d*x+c)*c*f^3*(d*x+c)+3*arccot(d*x+c)*d*e*
f^2*(d*x+c)+3/2*arccot(d*x+c)*ln(1+(d*x+c)^2)*c^2*f^3-3*arccot(d*x+c)*ln(1+(d*x+c)^2)*c*d*e*f^2+3/2*arccot(d*x
+c)*ln(1+(d*x+c)^2)*d^2*e^2*f-1/2*arccot(d*x+c)*ln(1+(d*x+c)^2)*f^3-arccot(d*x+c)*arctan(d*x+c)*c^3*f^3+3*arcc
ot(d*x+c)*arctan(d*x+c)*c^2*d*e*f^2-3*arccot(d*x+c)*arctan(d*x+c)*c*d^2*e^2*f+arccot(d*x+c)*arctan(d*x+c)*d^3*
e^3+3*arccot(d*x+c)*arctan(d*x+c)*c*f^3-3*arccot(d*x+c)*arctan(d*x+c)*d*e*f^2+1/2*f^2*(f*(d*x+c)+1/2*(-6*c*f+6
*d*e)*ln(1+(d*x+c)^2)-f*arctan(d*x+c))+1/2*f*(3*c^2*f^2-6*c*d*e*f+3*d^2*e^2-f^2)*(-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))))+1/4*(-2*c^3*f^3+6*c^2*d*e*
f^2-6*c*d^2*e^2*f+2*d^3*e^3+6*c*f^3-6*d*e*f^2)*arctan(d*x+c)^2))+2/3*a*b/f*arccot(d*x+c)*e^3-5/3/d^3*c^2*f^2*b
*a+1/3/d*f^2*b*a*x^2-1/3*a*b/d^3*f^2*ln(1+(d*x+c)^2)-2*a*b/d^2*f*ln(1+(d*x+c)^2)*c*e+2*b/d^2*arctan(d*x+c)*a*c
^2*e*f-2*b/d*arctan(d*x+c)*a*c*e^2+2*a*b*f*arccot(d*x+c)*e*x^2+2/d^2*c*f*e*b*a-4/3/d^2*c*f^2*x*b*a+2/d*e*x*f*b
*a+a*b/d^3*f^2*ln(1+(d*x+c)^2)*c^2+a*b/d*ln(1+(d*x+c)^2)*e^2-2/3*b/d^3*arctan(d*x+c)*a*c^3*f^2+2/3*a*b/f*arcta
n(d*x+c)*e^3+2*b/d^3*arctan(d*x+c)*a*c*f^2-2*b/d^2*arctan(d*x+c)*a*e*f+2/3*a*b*f^2*arccot(d*x+c)*x^3+2*a*b*arc
cot(d*x+c)*x*e^2

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/12*b^2*f^2*x^3*arctan2(1, d*x + c)^2 + 1/4*b^2*e*f*x^2*arctan2(1, d*x + c)^2 + 1/3*a^2*f^2*x^3 + 1/4*b^2*e^2
*x*arctan2(1, d*x + c)^2 + a^2*e*f*x^2 + 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*b*e*f + 1/3*(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*b*f^2 + a^2
*e^2*x + (2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a*b*e^2/d - 1/48*(b^2*f^2*x^3 + 3*b^2*e*f*x^2 +
3*b^2*e^2*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + integrate(1/48*(36*b^2*d^2*f^2*x^4*arctan2(1, d*x + c)^2 + 8
*(9*b^2*d^2*e*f*arctan2(1, d*x + c)^2 + (9*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*f^2)*x^3 +
36*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e^2 + 12*(3*b^2*d^2*e^2*arctan2(1, d*x + c)^2
+ 2*(6*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*e*f + 3*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*a
rctan2(1, d*x + c)^2)*f^2)*x^2 + 3*(b^2*d^2*f^2*x^4 + 2*(b^2*d^2*e*f + b^2*c*d*f^2)*x^3 + (b^2*c^2 + b^2)*e^2
+ (b^2*d^2*e^2 + 4*b^2*c*d*e*f + (b^2*c^2 + b^2)*f^2)*x^2 + 2*(b^2*c*d*e^2 + (b^2*c^2 + b^2)*e*f)*x)*log(d^2*x
^2 + 2*c*d*x + c^2 + 1)^2 + 24*((3*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*e^2 + 3*(b^2*c^2*a
rctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e*f)*x + 4*(b^2*d^2*f^2*x^4 + 3*b^2*c*d*e^2*x + (3*b^2*d^2*e
*f + b^2*c*d*f^2)*x^3 + 3*(b^2*d^2*e^2 + b^2*c*d*e*f)*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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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