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

Optimal. Leaf size=233 \[ \frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b f x \left (-\left (1-6 c^2\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac{b \left (-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (c^4-6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left ((c+d x)^2+1\right )}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4} \]

[Out]

(b*f*(6*d^2*e^2 - 12*c*d*e*f - (1 - 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*
(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcCot[c + d*x]))/(4*f) + (b*(d^4*e^4 - 4*c*d^3*e^3*f - 6*(1 - c^2
)*d^2*e^2*f^2 + 4*c*(3 - c^2)*d*e*f^3 + (1 - 6*c^2 + c^4)*f^4)*ArcTan[c + d*x])/(4*d^4*f) + (b*(d*e - c*f)*(d*
e + f - c*f)*(d*e - (1 + c)*f)*Log[1 + (c + d*x)^2])/(2*d^4)

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Rubi [A]  time = 0.356664, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5048, 4863, 702, 635, 203, 260} \[ \frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b f x \left (-\left (1-6 c^2\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac{b \left (-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (c^4-6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left ((c+d x)^2+1\right )}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f*(6*d^2*e^2 - 12*c*d*e*f - (1 - 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*
(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcCot[c + d*x]))/(4*f) + (b*(d^4*e^4 - 4*c*d^3*e^3*f - 6*(1 - c^2
)*d^2*e^2*f^2 + 4*c*(3 - c^2)*d*e*f^3 + (1 - 6*c^2 + c^4)*f^4)*ArcTan[c + d*x])/(4*d^4*f) + (b*(d*e - c*f)*(d*
e + f - c*f)*(d*e - (1 + c)*f)*Log[1 + (c + d*x)^2])/(2*d^4)

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 4863

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

Rule 702

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

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

Rubi steps

\begin{align*} \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^3 \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^4}{1+x^2} \, dx,x,c+d x\right )}{4 f}\\ &=\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \operatorname{Subst}\left (\int \left (\frac{f^2 \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right )}{d^4}+\frac{4 f^3 (d e-c f) x}{d^4}+\frac{f^4 x^2}{d^4}+\frac{d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{d^4 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{4 f}\\ &=\frac{b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac{b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}+\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \operatorname{Subst}\left (\int \frac{d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f}\\ &=\frac{b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac{b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}+\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{(b (d e-c f) (d e+f-c f) (d e-(1+c) f)) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d^4}+\frac{\left (b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f}\\ &=\frac{b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac{b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}+\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right ) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b (d e-c f) (d e+f-c f) (d e-(1+c) f) \log \left (1+(c+d x)^2\right )}{2 d^4}\\ \end{align*}

Mathematica [C]  time = 0.266223, size = 157, normalized size = 0.67 \[ \frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )+\frac{b \left (6 d f^2 x \left (\left (6 c^2-1\right ) f^2-12 c d e f+6 d^2 e^2\right )+12 f^3 (c+d x)^2 (d e-c f)-3 i (d e-(c-i) f)^4 \log (-c-d x+i)+3 i (d e-(c+i) f)^4 \log (c+d x+i)+2 f^4 (c+d x)^3\right )}{6 d^4}}{4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)^4*(a + b*ArcCot[c + d*x]) + (b*(6*d*f^2*(6*d^2*e^2 - 12*c*d*e*f + (-1 + 6*c^2)*f^2)*x + 12*f^3*(d*e
- c*f)*(c + d*x)^2 + 2*f^4*(c + d*x)^3 - (3*I)*(d*e - (-I + c)*f)^4*Log[I - c - d*x] + (3*I)*(d*e - (I + c)*f
)^4*Log[I + c + d*x]))/(6*d^4))/(4*f)

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Maple [B]  time = 0.052, size = 526, normalized size = 2.3 \begin{align*}{\frac{b{f}^{3}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) c}{2\,{d}^{4}}}-{\frac{3\,bf\arctan \left ( dx+c \right ){e}^{2}}{2\,{d}^{2}}}-{\frac{3\,b{f}^{3}\arctan \left ( dx+c \right ){c}^{2}}{2\,{d}^{4}}}-{\frac{b\arctan \left ( dx+c \right ) c{e}^{3}}{d}}+{\frac{3\,bf{\rm arccot} \left (dx+c\right ){e}^{2}{x}^{2}}{2}}+{\frac{3\,b{f}^{3}{c}^{2}x}{4\,{d}^{3}}}+{\frac{3\,bf{e}^{2}x}{2\,d}}-{\frac{b{f}^{3}c{x}^{2}}{4\,{d}^{2}}}+{\frac{b{f}^{2}e{x}^{2}}{2\,d}}-{\frac{b{f}^{3}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){c}^{3}}{2\,{d}^{4}}}+{\frac{b{f}^{3}\arctan \left ( dx+c \right ){c}^{4}}{4\,{d}^{4}}}-{\frac{b{f}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) e}{2\,{d}^{3}}}+b{f}^{2}{\rm arccot} \left (dx+c\right )e{x}^{3}+{\frac{3\,bfc{e}^{2}}{2\,{d}^{2}}}-{\frac{5\,b{f}^{2}{c}^{2}e}{2\,{d}^{3}}}+ax{e}^{3}+{\frac{a{f}^{3}{x}^{4}}{4}}+{\frac{13\,b{f}^{3}{c}^{3}}{12\,{d}^{4}}}+{\frac{a{e}^{4}}{4\,f}}-{\frac{b{f}^{3}c}{4\,{d}^{4}}}-{\frac{b{f}^{3}x}{4\,{d}^{3}}}+a{f}^{2}{x}^{3}e+{\frac{3\,af{x}^{2}{e}^{2}}{2}}+{\frac{b{f}^{3}{x}^{3}}{12\,d}}+{\frac{b{\rm arccot} \left (dx+c\right ){e}^{4}}{4\,f}}+{\frac{b{f}^{3}{\rm arccot} \left (dx+c\right ){x}^{4}}{4}}+{\frac{b\arctan \left ( dx+c \right ){e}^{4}}{4\,f}}+{\rm arccot} \left (dx+c\right )xb{e}^{3}+{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){e}^{3}}{2\,d}}+{\frac{b{f}^{3}\arctan \left ( dx+c \right ) }{4\,{d}^{4}}}+3\,{\frac{b{f}^{2}\arctan \left ( dx+c \right ) ce}{{d}^{3}}}+{\frac{3\,b{f}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){c}^{2}e}{2\,{d}^{3}}}-{\frac{3\,bf\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) c{e}^{2}}{2\,{d}^{2}}}+{\frac{3\,bf\arctan \left ( dx+c \right ){c}^{2}{e}^{2}}{2\,{d}^{2}}}-{\frac{b{f}^{2}\arctan \left ( dx+c \right ){c}^{3}e}{{d}^{3}}}-2\,{\frac{b{f}^{2}cex}{{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d^4*b*f^3*ln(1+(d*x+c)^2)*c-3/2/d^2*b*f*arctan(d*x+c)*e^2-3/2/d^4*b*f^3*arctan(d*x+c)*c^2-1/d*b*arctan(d*x
+c)*c*e^3+3/2*b*f*arccot(d*x+c)*e^2*x^2+3/4*b/d^3*f^3*c^2*x+3/2*b/d*f*e^2*x-1/4/d^2*b*f^3*c*x^2+1/2/d*b*f^2*e*
x^2-1/2/d^4*b*f^3*ln(1+(d*x+c)^2)*c^3+1/4/d^4*b*f^3*arctan(d*x+c)*c^4-1/2/d^3*b*f^2*ln(1+(d*x+c)^2)*e+b*f^2*ar
ccot(d*x+c)*e*x^3+3/2/d^2*b*f*c*e^2-5/2/d^3*b*f^2*c^2*e+a*x*e^3+1/4*a*f^3*x^4+13/12/d^4*b*f^3*c^3+1/4*a/f*e^4-
1/4/d^4*b*f^3*c-1/4*b/d^3*f^3*x+a*f^2*x^3*e+3/2*a*f*x^2*e^2+1/12/d*b*f^3*x^3+1/4*b/f*arccot(d*x+c)*e^4+1/4*b*f
^3*arccot(d*x+c)*x^4+1/4*b/f*arctan(d*x+c)*e^4+arccot(d*x+c)*x*b*e^3+1/2/d*b*ln(1+(d*x+c)^2)*e^3+1/4/d^4*b*f^3
*arctan(d*x+c)+3/d^3*b*f^2*arctan(d*x+c)*c*e+3/2/d^3*b*f^2*ln(1+(d*x+c)^2)*c^2*e-3/2/d^2*b*f*ln(1+(d*x+c)^2)*c
*e^2+3/2/d^2*b*f*arctan(d*x+c)*c^2*e^2-1/d^3*b*f^2*arctan(d*x+c)*c^3*e-2*b/d^2*f^2*c*e*x

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Maxima [A]  time = 1.49634, size = 460, normalized size = 1.97 \begin{align*} \frac{1}{4} \, a f^{3} x^{4} + a e f^{2} x^{3} + \frac{3}{2} \, a e^{2} f x^{2} + \frac{3}{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 )} b e^{2} f + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{arccot}\left (d x + c\right ) + d{\left (\frac{d x^{2} - 4 \, c x}{d^{3}} - \frac{2 \,{\left (c^{3} - 3 \, c\right )} \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{4}} + \frac{{\left (3 \, c^{2} - 1\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4}}\right )}\right )} b e f^{2} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arccot}\left (d x + c\right ) + d{\left (\frac{d^{2} x^{3} - 3 \, c d x^{2} + 3 \,{\left (3 \, c^{2} - 1\right )} x}{d^{4}} + \frac{3 \,{\left (c^{4} - 6 \, c^{2} + 1\right )} \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{5}} - \frac{6 \,{\left (c^{3} - c\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{5}}\right )}\right )} b f^{3} + a e^{3} 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 )} b e^{3}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*f^3*x^4 + a*e*f^2*x^3 + 3/2*a*e^2*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))*b*e^2*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))*b*e*
f^2 + 1/12*(3*x^4*arccot(d*x + c) + d*((d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 - 1)*x)/d^4 + 3*(c^4 - 6*c^2 + 1)*arcta
n((d^2*x + c*d)/d)/d^5 - 6*(c^3 - c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^5))*b*f^3 + a*e^3*x + 1/2*(2*(d*x + c)
*arccot(d*x + c) + log((d*x + c)^2 + 1))*b*e^3/d

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Fricas [A]  time = 2.46891, size = 694, normalized size = 2.98 \begin{align*} \frac{3 \, a d^{4} f^{3} x^{4} +{\left (12 \, a d^{4} e f^{2} + b d^{3} f^{3}\right )} x^{3} + 3 \,{\left (6 \, a d^{4} e^{2} f + 2 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 3 \,{\left (4 \, a d^{4} e^{3} + 6 \, b d^{3} e^{2} f - 8 \, b c d^{2} e f^{2} +{\left (3 \, b c^{2} - b\right )} d f^{3}\right )} x + 3 \,{\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} e f^{2} x^{3} + 6 \, b d^{4} e^{2} f x^{2} + 4 \, b d^{4} e^{3} x\right )} \operatorname{arccot}\left (d x + c\right ) - 3 \,{\left (4 \, b c d^{3} e^{3} - 6 \,{\left (b c^{2} - b\right )} d^{2} e^{2} f + 4 \,{\left (b c^{3} - 3 \, b c\right )} d e f^{2} -{\left (b c^{4} - 6 \, b c^{2} + b\right )} f^{3}\right )} \arctan \left (d x + c\right ) + 6 \,{\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f +{\left (3 \, b c^{2} - b\right )} d e f^{2} -{\left (b c^{3} - b c\right )} f^{3}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{12 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a*d^4*f^3*x^4 + (12*a*d^4*e*f^2 + b*d^3*f^3)*x^3 + 3*(6*a*d^4*e^2*f + 2*b*d^3*e*f^2 - b*c*d^2*f^3)*x^2
+ 3*(4*a*d^4*e^3 + 6*b*d^3*e^2*f - 8*b*c*d^2*e*f^2 + (3*b*c^2 - b)*d*f^3)*x + 3*(b*d^4*f^3*x^4 + 4*b*d^4*e*f^
2*x^3 + 6*b*d^4*e^2*f*x^2 + 4*b*d^4*e^3*x)*arccot(d*x + c) - 3*(4*b*c*d^3*e^3 - 6*(b*c^2 - b)*d^2*e^2*f + 4*(b
*c^3 - 3*b*c)*d*e*f^2 - (b*c^4 - 6*b*c^2 + b)*f^3)*arctan(d*x + c) + 6*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + (3*b*c^2
- b)*d*e*f^2 - (b*c^3 - b*c)*f^3)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/d^4

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Sympy [A]  time = 8.92132, size = 627, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*e**3*x + 3*a*e**2*f*x**2/2 + a*e*f**2*x**3 + a*f**3*x**4/4 - b*c**4*f**3*acot(c + d*x)/(4*d**4) +
b*c**3*e*f**2*acot(c + d*x)/d**3 - b*c**3*f**3*log(c**2/d**2 + 2*c*x/d + x**2 + d**(-2))/(2*d**4) - 3*b*c**2*
e**2*f*acot(c + d*x)/(2*d**2) + 3*b*c**2*e*f**2*log(c**2/d**2 + 2*c*x/d + x**2 + d**(-2))/(2*d**3) + 3*b*c**2*
f**3*x/(4*d**3) + 3*b*c**2*f**3*acot(c + d*x)/(2*d**4) + b*c*e**3*acot(c + d*x)/d - 3*b*c*e**2*f*log(c**2/d**2
+ 2*c*x/d + x**2 + d**(-2))/(2*d**2) - 2*b*c*e*f**2*x/d**2 - b*c*f**3*x**2/(4*d**2) - 3*b*c*e*f**2*acot(c + d
*x)/d**3 + b*c*f**3*log(c**2/d**2 + 2*c*x/d + x**2 + d**(-2))/(2*d**4) + b*e**3*x*acot(c + d*x) + 3*b*e**2*f*x
**2*acot(c + d*x)/2 + b*e*f**2*x**3*acot(c + d*x) + b*f**3*x**4*acot(c + d*x)/4 + b*e**3*log(c**2/d**2 + 2*c*x
/d + x**2 + d**(-2))/(2*d) + 3*b*e**2*f*x/(2*d) + b*e*f**2*x**2/(2*d) + b*f**3*x**3/(12*d) + 3*b*e**2*f*acot(c
+ d*x)/(2*d**2) - b*e*f**2*log(c**2/d**2 + 2*c*x/d + x**2 + d**(-2))/(2*d**3) - b*f**3*x/(4*d**3) - b*f**3*ac
ot(c + d*x)/(4*d**4), Ne(d, 0)), ((a + b*acot(c))*(e**3*x + 3*e**2*f*x**2/2 + e*f**2*x**3 + f**3*x**4/4), True
))

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Giac [B]  time = 3.22393, size = 1045, normalized size = 4.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*pi*b*d^4*f^3*x^4*floor(1/2*(pi*sgn(d*x + c) - 2*arctan(1/(d*x + c)))/pi) - 3*pi*b*d^4*f^3*x^4*sgn(d*x
+ c) + 3*pi*b*d^4*f^3*x^4 + 6*b*d^4*f^3*x^4*arctan(1/(d*x + c)) + 6*a*d^4*f^3*x^4 + 24*b*d^4*f^2*x^3*arctan(1/
(d*x + c))*e + 24*a*d^4*f^2*x^3*e + 2*b*d^3*f^3*x^3 + 36*b*d^4*f*x^2*arctan(1/(d*x + c))*e^2 + 3*pi*b*c^4*f^3*
sgn(d*x + c) - 12*pi*b*c^3*d*f^2*e*sgn(d*x + c) - 3*pi*b*c^4*f^3 - 6*b*c*d^2*f^3*x^2 - 6*b*c^4*f^3*arctan(1/(d
*x + c)) + 36*a*d^4*f*x^2*e^2 + 12*pi*b*c^3*d*f^2*e + 12*b*d^3*f^2*x^2*e + 24*b*c^3*d*f^2*arctan(1/(d*x + c))*
e + 18*pi*b*c^2*d^2*f*e^2*sgn(d*x + c) + 18*b*c^2*d*f^3*x + 24*b*d^4*x*arctan(1/(d*x + c))*e^3 - 18*pi*b*c^2*d
^2*f*e^2 - 36*b*c^2*d^2*f*arctan(1/(d*x + c))*e^2 - 48*b*c*d^2*f^2*x*e - 12*b*c^3*f^3*log(d^2*x^2 + 2*c*d*x +
c^2 + 1) + 36*b*c^2*d*f^2*e*log(d^2*x^2 + 2*c*d*x + c^2 + 1) - 18*pi*b*c^2*f^3*sgn(d*x + c) + 36*pi*b*c*d*f^2*
e*sgn(d*x + c) + 18*pi*b*c^2*f^3 + 36*b*c^2*f^3*arctan(1/(d*x + c)) + 24*a*d^4*x*e^3 - 24*b*c*d^3*arctan(d*x +
c)*e^3 + 36*b*d^3*f*x*e^2 - 36*pi*b*c*d*f^2*e - 72*b*c*d*f^2*arctan(1/(d*x + c))*e - 36*b*c*d^2*f*e^2*log(d^2
*x^2 + 2*c*d*x + c^2 + 1) - 18*pi*b*d^2*f*e^2*sgn(d*x + c) - 6*b*d*f^3*x + 18*pi*b*d^2*f*e^2 + 36*b*d^2*f*arct
an(1/(d*x + c))*e^2 + 12*b*c*f^3*log(d^2*x^2 + 2*c*d*x + c^2 + 1) + 12*b*d^3*e^3*log(d^2*x^2 + 2*c*d*x + c^2 +
1) - 12*b*d*f^2*e*log(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*pi*b*f^3*sgn(d*x + c) - 3*pi*b*f^3 - 6*b*f^3*arctan(1/
(d*x + c)))/d^4