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

Optimal. Leaf size=220 \[ \frac{i b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}-\frac{2 b (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{a b f x}{d}+\frac{b^2 f \log \left ((c+d x)^2+1\right )}{2 d^2}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.384159, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5048, 4865, 4847, 260, 4985, 4885, 4921, 4855, 2402, 2315} \[ \frac{i b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}-\frac{2 b (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{a b f x}{d}+\frac{b^2 f \log \left ((c+d x)^2+1\right )}{2 d^2}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5048

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 4865

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 4847

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

Rule 260

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 4985

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 4885

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 4921

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 4855

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])^p*Lo
g[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 2402

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 2315

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

Rubi steps

\begin{align*} \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right ) \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{b \operatorname{Subst}\left (\int \left (\frac{f^2 \left (a+b \cot ^{-1}(x)\right )}{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 )}{d^2 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{b \operatorname{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 )}{1+x^2} \, dx,x,c+d x\right )}{d^2 f}+\frac{(b f) \operatorname{Subst}\left (\int \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{a b f x}{d}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{b \operatorname{Subst}\left (\int \left (\frac{(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(x)\right )}{1+x^2}-\frac{2 f (-d e+c f) x \left (a+b \cot ^{-1}(x)\right )}{1+x^2}\right ) \, dx,x,c+d x\right )}{d^2 f}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \cot ^{-1}(x) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{a b f x}{d}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d^2}+\frac{(2 b (d e-c f)) \operatorname{Subst}\left (\int \frac{x \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}+\frac{(b (d e+f-c f) (d e-(1+c) f)) \operatorname{Subst}\left (\int \frac{a+b \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{d^2 f}\\ &=\frac{a b f x}{d}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac{(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac{b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}-\frac{(2 b (d e-c f)) \operatorname{Subst}\left (\int \frac{a+b \cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{a b f x}{d}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac{(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac{2 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}-\frac{\left (2 b^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{a b f x}{d}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac{(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac{2 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac{\left (2 i b^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i (c+d x)}\right )}{d^2}\\ &=\frac{a b f x}{d}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac{(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac{2 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac{i b^2 (d e-c f) \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.561316, size = 286, normalized size = 1.3 \[ \frac{2 i b^2 (d e-c f) \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )-a^2 c^2 f+2 a^2 c d e+2 a^2 d^2 e x+a^2 d^2 f x^2+2 b \cot ^{-1}(c+d x) \left (-(c+d x) (a c f-a d (2 e+f x)-b f)-2 b (d e-c f) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )-4 a b d e \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )+4 a b c f \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )-2 a b f \tan ^{-1}(c+d x)+2 a b c f+2 a b d f x+b^2 (c+d x+i) \cot ^{-1}(c+d x)^2 (d (2 e+f x)-(c+i) f)-2 b^2 f \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )}{2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.135, size = 766, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I/d^2*b^2*dilog(1/2*I*(d*x+c-I))*c*f+1/4*I/d^2*b^2*ln(d*x+c+I)^2*c*f-1/2*I/d*b^2*ln(d*x+c-I)*ln(1+(d*x+c)^
2)*e-1/2*I/d*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*e+1/2*I/d*b^2*ln(d*x+c+I)*ln(1+(d*x+c)^2)*e+1/2*I/d*b^2*ln(d*
x+c-I)*ln(-1/2*I*(d*x+c+I))*e-1/d^2*a*b*arccot(d*x+c)*c^2*f-1/d^2*b^2*ln(1+(d*x+c)^2)*arccot(d*x+c)*c*f-1/d^2*
a*b*ln(1+(d*x+c)^2)*c*f-1/4*I/d^2*b^2*ln(d*x+c-I)^2*c*f-1/2*I/d^2*b^2*dilog(-1/2*I*(d*x+c+I))*c*f+2/d*arccot(d
*x+c)*a*b*c*e+1/2*a^2*x^2*f+a^2*x*e+1/2*b^2*f*ln(1+(d*x+c)^2)/d^2-1/2/d^2*b^2*f*arctan(d*x+c)^2+1/2*b^2*arccot
(d*x+c)^2*f*x^2+arccot(d*x+c)^2*x*b^2*e-1/2/d^2*a^2*f*c^2+1/d*a^2*c*e+2*arccot(d*x+c)*x*a*b*e-1/d^2*b^2*arctan
(d*x+c)*arccot(d*x+c)*f+1/d*a*b*ln(1+(d*x+c)^2)*e-1/d^2*a*b*f*arctan(d*x+c)+1/d*b^2*ln(1+(d*x+c)^2)*arccot(d*x
+c)*e+1/d*arccot(d*x+c)^2*b^2*c*e+1/d*b^2*arccot(d*x+c)*f*x+1/d^2*b^2*arccot(d*x+c)*f*c-1/2/d^2*b^2*arccot(d*x
+c)^2*f*c^2-1/4*I/d*b^2*ln(d*x+c+I)^2*e-1/2*I/d*b^2*dilog(1/2*I*(d*x+c-I))*e+1/4*I/d*b^2*ln(d*x+c-I)^2*e+1/2*I
/d*b^2*dilog(-1/2*I*(d*x+c+I))*e+a*b*arccot(d*x+c)*f*x^2+1/d^2*a*b*f*c+1/2*I/d^2*b^2*ln(d*x+c-I)*ln(1+(d*x+c)^
2)*c*f+1/2*I/d^2*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*c*f-1/2*I/d^2*b^2*ln(d*x+c+I)*ln(1+(d*x+c)^2)*c*f-1/2*I/d
^2*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*c*f+a*b*f*x/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, b^{2} f x^{2} \arctan \left (1, d x + c\right )^{2} + \frac{1}{4} \, b^{2} e x \arctan \left (1, d x + c\right )^{2} + \frac{1}{2} \, a^{2} f x^{2} +{\left (x^{2} \operatorname{arccot}\left (d x + c\right ) + d{\left (\frac{x}{d^{2}} + \frac{{\left (c^{2} - 1\right )} \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{3}} - \frac{c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} a b f + a^{2} e x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} a b e}{d} - \frac{1}{32} \,{\left (b^{2} f x^{2} + 2 \, b^{2} e x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + \int \frac{12 \, b^{2} d^{2} f x^{3} \arctan \left (1, d x + c\right )^{2} + 4 \,{\left (3 \, b^{2} d^{2} e \arctan \left (1, d x + c\right )^{2} +{\left (6 \, b^{2} c \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )\right )} d f\right )} x^{2} +{\left (b^{2} d^{2} f x^{3} +{\left (b^{2} d^{2} e + 2 \, b^{2} c d f\right )} x^{2} +{\left (b^{2} c^{2} + b^{2}\right )} e +{\left (2 \, b^{2} c d e +{\left (b^{2} c^{2} + b^{2}\right )} f\right )} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 12 \,{\left (b^{2} c^{2} \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )^{2}\right )} e + 4 \,{\left (2 \,{\left (3 \, b^{2} c \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )\right )} d e + 3 \,{\left (b^{2} c^{2} \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )^{2}\right )} f\right )} x + 2 \,{\left (b^{2} d^{2} f x^{3} + 2 \, b^{2} c d e x +{\left (2 \, b^{2} d^{2} e + b^{2} c d f\right )} x^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{16 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*f*x^2*arctan2(1, d*x + c)^2 + 1/4*b^2*e*x*arctan2(1, d*x + c)^2 + 1/2*a^2*f*x^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*f + a^2*e*x
+ (2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a*b*e/d - 1/32*(b^2*f*x^2 + 2*b^2*e*x)*log(d^2*x^2 + 2
*c*d*x + c^2 + 1)^2 + integrate(1/16*(12*b^2*d^2*f*x^3*arctan2(1, d*x + c)^2 + 4*(3*b^2*d^2*e*arctan2(1, d*x +
c)^2 + (6*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*f)*x^2 + (b^2*d^2*f*x^3 + (b^2*d^2*e + 2*b
^2*c*d*f)*x^2 + (b^2*c^2 + b^2)*e + (2*b^2*c*d*e + (b^2*c^2 + b^2)*f)*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 +
12*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e + 4*(2*(3*b^2*c*arctan2(1, d*x + c)^2 + b^2*a
rctan2(1, d*x + c))*d*e + 3*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*f)*x + 2*(b^2*d^2*f*x^
3 + 2*b^2*c*d*e*x + (2*b^2*d^2*e + b^2*c*d*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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} f x + a^{2} e +{\left (b^{2} f x + b^{2} e\right )} \operatorname{arccot}\left (d x + c\right )^{2} + 2 \,{\left (a b f x + a b e\right )} \operatorname{arccot}\left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acot}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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