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

Optimal. Leaf size=154 \[ \frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )}{3 f}+\frac{b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left ((c+d x)^2+1\right )}{6 d^3}+\frac{b (d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \tan ^{-1}(c+d x)}{3 d^3 f}+\frac{b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2}{6 d^3} \]

[Out]

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

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Rubi [A]  time = 0.185669, antiderivative size = 154, 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} \[ \frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )}{3 f}+\frac{b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left ((c+d x)^2+1\right )}{6 d^3}+\frac{b (d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \tan ^{-1}(c+d x)}{3 d^3 f}+\frac{b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2}{6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.145346, size = 118, normalized size = 0.77 \[ \frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )+\frac{b \left (6 d f^2 x (d e-c f)-i (d e-(c-i) f)^3 \log (-c-d x+i)+i (d e-(c+i) f)^3 \log (c+d x+i)+f^3 (c+d x)^2\right )}{2 d^3}}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.048, size = 312, normalized size = 2. \begin{align*}{\frac{bf\arctan \left ( dx+c \right ){c}^{2}e}{{d}^{2}}}-{\frac{bf\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) ce}{{d}^{2}}}-{\frac{b\arctan \left ( dx+c \right ) c{e}^{2}}{d}}-{\frac{b{f}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{6\,{d}^{3}}}-{\frac{5\,b{f}^{2}{c}^{2}}{6\,{d}^{3}}}+{\rm arccot} \left (dx+c\right )xb{e}^{2}+{\frac{b{f}^{2}{\rm arccot} \left (dx+c\right ){x}^{3}}{3}}-{\frac{bf\arctan \left ( dx+c \right ) e}{{d}^{2}}}+{\frac{b{f}^{2}\arctan \left ( dx+c \right ) c}{{d}^{3}}}+{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){e}^{2}}{2\,d}}-{\frac{b{f}^{2}\arctan \left ( dx+c \right ){c}^{3}}{3\,{d}^{3}}}+{\frac{b\arctan \left ( dx+c \right ){e}^{3}}{3\,f}}+{\frac{b{f}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){c}^{2}}{2\,{d}^{3}}}+{\frac{bfex}{d}}-{\frac{2\,b{f}^{2}cx}{3\,{d}^{2}}}+{\frac{b{\rm arccot} \left (dx+c\right ){e}^{3}}{3\,f}}+af{x}^{2}e+ax{e}^{2}+{\frac{bfce}{{d}^{2}}}+{\frac{b{f}^{2}{x}^{2}}{6\,d}}+{\frac{a{f}^{2}{x}^{3}}{3}}+{\frac{a{e}^{3}}{3\,f}}+bf{\rm arccot} \left (dx+c\right )e{x}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*b*f*arctan(d*x+c)*c^2*e-1/d^2*b*f*ln(1+(d*x+c)^2)*c*e-1/d*b*arctan(d*x+c)*c*e^2-1/6/d^3*b*f^2*ln(1+(d*x+
c)^2)-5/6/d^3*b*f^2*c^2+arccot(d*x+c)*x*b*e^2+1/3*b*f^2*arccot(d*x+c)*x^3-1/d^2*b*f*arctan(d*x+c)*e+1/d^3*b*f^
2*arctan(d*x+c)*c+1/2/d*b*ln(1+(d*x+c)^2)*e^2-1/3/d^3*b*f^2*arctan(d*x+c)*c^3+1/3*b/f*arctan(d*x+c)*e^3+1/2/d^
3*b*f^2*ln(1+(d*x+c)^2)*c^2+b/d*f*e*x-2/3*b/d^2*f^2*c*x+1/3*b/f*arccot(d*x+c)*e^3+a*f*x^2*e+a*x*e^2+1/d^2*b*f*
c*e+1/6/d*b*f^2*x^2+1/3*a*f^2*x^3+1/3*a/f*e^3+b*f*arccot(d*x+c)*e*x^2

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Maxima [A]  time = 1.49671, size = 292, normalized size = 1.9 \begin{align*} \frac{1}{3} \, a f^{2} x^{3} + a e 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 )} b e f + \frac{1}{6} \,{\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 f^{2} + a e^{2} 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^{2}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.38821, size = 459, normalized size = 2.98 \begin{align*} \frac{2 \, a d^{3} f^{2} x^{3} +{\left (6 \, a d^{3} e f + b d^{2} f^{2}\right )} x^{2} + 2 \,{\left (3 \, a d^{3} e^{2} + 3 \, b d^{2} e f - 2 \, b c d f^{2}\right )} x + 2 \,{\left (b d^{3} f^{2} x^{3} + 3 \, b d^{3} e f x^{2} + 3 \, b d^{3} e^{2} x\right )} \operatorname{arccot}\left (d x + c\right ) - 2 \,{\left (3 \, b c d^{2} e^{2} - 3 \,{\left (b c^{2} - b\right )} d e f +{\left (b c^{3} - 3 \, b c\right )} f^{2}\right )} \arctan \left (d x + c\right ) +{\left (3 \, b d^{2} e^{2} - 6 \, b c d e f +{\left (3 \, b c^{2} - b\right )} f^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{6 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 4.35104, size = 357, normalized size = 2.32 \begin{align*} \begin{cases} a e^{2} x + a e f x^{2} + \frac{a f^{2} x^{3}}{3} + \frac{b c^{3} f^{2} \operatorname{acot}{\left (c + d x \right )}}{3 d^{3}} - \frac{b c^{2} e f \operatorname{acot}{\left (c + d x \right )}}{d^{2}} + \frac{b c^{2} f^{2} \log{\left (\frac{c^{2}}{d^{2}} + \frac{2 c x}{d} + x^{2} + \frac{1}{d^{2}} \right )}}{2 d^{3}} + \frac{b c e^{2} \operatorname{acot}{\left (c + d x \right )}}{d} - \frac{b c e f \log{\left (\frac{c^{2}}{d^{2}} + \frac{2 c x}{d} + x^{2} + \frac{1}{d^{2}} \right )}}{d^{2}} - \frac{2 b c f^{2} x}{3 d^{2}} - \frac{b c f^{2} \operatorname{acot}{\left (c + d x \right )}}{d^{3}} + b e^{2} x \operatorname{acot}{\left (c + d x \right )} + b e f x^{2} \operatorname{acot}{\left (c + d x \right )} + \frac{b f^{2} x^{3} \operatorname{acot}{\left (c + d x \right )}}{3} + \frac{b e^{2} \log{\left (\frac{c^{2}}{d^{2}} + \frac{2 c x}{d} + x^{2} + \frac{1}{d^{2}} \right )}}{2 d} + \frac{b e f x}{d} + \frac{b f^{2} x^{2}}{6 d} + \frac{b e f \operatorname{acot}{\left (c + d x \right )}}{d^{2}} - \frac{b f^{2} \log{\left (\frac{c^{2}}{d^{2}} + \frac{2 c x}{d} + x^{2} + \frac{1}{d^{2}} \right )}}{6 d^{3}} & \text{for}\: d \neq 0 \\\left (a + b \operatorname{acot}{\left (c \right )}\right ) \left (e^{2} x + e f x^{2} + \frac{f^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*e**2*x + a*e*f*x**2 + a*f**2*x**3/3 + b*c**3*f**2*acot(c + d*x)/(3*d**3) - b*c**2*e*f*acot(c + d*
x)/d**2 + b*c**2*f**2*log(c**2/d**2 + 2*c*x/d + x**2 + d**(-2))/(2*d**3) + b*c*e**2*acot(c + d*x)/d - b*c*e*f*
log(c**2/d**2 + 2*c*x/d + x**2 + d**(-2))/d**2 - 2*b*c*f**2*x/(3*d**2) - b*c*f**2*acot(c + d*x)/d**3 + b*e**2*
x*acot(c + d*x) + b*e*f*x**2*acot(c + d*x) + b*f**2*x**3*acot(c + d*x)/3 + b*e**2*log(c**2/d**2 + 2*c*x/d + x*
*2 + d**(-2))/(2*d) + b*e*f*x/d + b*f**2*x**2/(6*d) + b*e*f*acot(c + d*x)/d**2 - b*f**2*log(c**2/d**2 + 2*c*x/
d + x**2 + d**(-2))/(6*d**3), Ne(d, 0)), ((a + b*acot(c))*(e**2*x + e*f*x**2 + f**2*x**3/3), True))

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Giac [B]  time = 1.4938, size = 641, normalized size = 4.16 \begin{align*} \frac{2 \, \pi b d^{3} f^{2} x^{3} \left \lfloor \frac{\pi \mathrm{sgn}\left (d x + c\right ) - 2 \, \arctan \left (\frac{1}{d x + c}\right )}{2 \, \pi } \right \rfloor - \pi b d^{3} f^{2} x^{3} \mathrm{sgn}\left (d x + c\right ) + \pi b d^{3} f^{2} x^{3} + 2 \, b d^{3} f^{2} x^{3} \arctan \left (\frac{1}{d x + c}\right ) + 2 \, a d^{3} f^{2} x^{3} + 6 \, b d^{3} f x^{2} \arctan \left (\frac{1}{d x + c}\right ) e + 6 \, a d^{3} f x^{2} e - \pi b c^{3} f^{2} \mathrm{sgn}\left (d x + c\right ) + 3 \, \pi b c^{2} d f e \mathrm{sgn}\left (d x + c\right ) + \pi b c^{3} f^{2} + b d^{2} f^{2} x^{2} + 2 \, b c^{3} f^{2} \arctan \left (\frac{1}{d x + c}\right ) + 6 \, b d^{3} x \arctan \left (\frac{1}{d x + c}\right ) e^{2} - 3 \, \pi b c^{2} d f e - 6 \, b c^{2} d f \arctan \left (\frac{1}{d x + c}\right ) e - 4 \, b c d f^{2} x + 6 \, a d^{3} x e^{2} - 6 \, b c d^{2} \arctan \left (d x + c\right ) e^{2} + 6 \, b d^{2} f x e + 3 \, b c^{2} f^{2} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 6 \, b c d f e \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 3 \, \pi b c f^{2} \mathrm{sgn}\left (d x + c\right ) - 3 \, \pi b d f e \mathrm{sgn}\left (d x + c\right ) - 3 \, \pi b c f^{2} - 6 \, b c f^{2} \arctan \left (\frac{1}{d x + c}\right ) + 3 \, \pi b d f e + 6 \, b d f \arctan \left (\frac{1}{d x + c}\right ) e + 3 \, b d^{2} e^{2} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - b f^{2} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{6 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*pi*b*d^3*f^2*x^3*floor(1/2*(pi*sgn(d*x + c) - 2*arctan(1/(d*x + c)))/pi) - pi*b*d^3*f^2*x^3*sgn(d*x + c
) + pi*b*d^3*f^2*x^3 + 2*b*d^3*f^2*x^3*arctan(1/(d*x + c)) + 2*a*d^3*f^2*x^3 + 6*b*d^3*f*x^2*arctan(1/(d*x + c
))*e + 6*a*d^3*f*x^2*e - pi*b*c^3*f^2*sgn(d*x + c) + 3*pi*b*c^2*d*f*e*sgn(d*x + c) + pi*b*c^3*f^2 + b*d^2*f^2*
x^2 + 2*b*c^3*f^2*arctan(1/(d*x + c)) + 6*b*d^3*x*arctan(1/(d*x + c))*e^2 - 3*pi*b*c^2*d*f*e - 6*b*c^2*d*f*arc
tan(1/(d*x + c))*e - 4*b*c*d*f^2*x + 6*a*d^3*x*e^2 - 6*b*c*d^2*arctan(d*x + c)*e^2 + 6*b*d^2*f*x*e + 3*b*c^2*f
^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1) - 6*b*c*d*f*e*log(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*pi*b*c*f^2*sgn(d*x + c)
- 3*pi*b*d*f*e*sgn(d*x + c) - 3*pi*b*c*f^2 - 6*b*c*f^2*arctan(1/(d*x + c)) + 3*pi*b*d*f*e + 6*b*d*f*arctan(1/
(d*x + c))*e + 3*b*d^2*e^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1) - b*f^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/d^3