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

Optimal. Leaf size=97 \[ \frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{2 f}+\frac{b (d e-c f) \log \left ((c+d x)^2+1\right )}{2 d^2}+\frac{b (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 d^2 f}+\frac{b f x}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.114413, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5048, 4863, 702, 635, 203, 260} \[ \frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{2 f}+\frac{b (d e-c f) \log \left ((c+d x)^2+1\right )}{2 d^2}+\frac{b (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 d^2 f}+\frac{b f x}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.0817701, size = 163, normalized size = 1.68 \[ a e x+\frac{1}{2} a f x^2+\frac{b e \left (\log \left (c^2+2 c d x+d^2 x^2+1\right )-2 c \tan ^{-1}(c+d x)\right )}{2 d}+\frac{b f \left (\frac{1}{2} d \left (\frac{c+d x}{d}-\frac{c}{d}\right )^2 \cot ^{-1}(c+d x)+\frac{1}{2} d \left (-\frac{i (-c+i)^2 \log (-c-d x+i)}{2 d^2}+\frac{i (c+i)^2 \log (c+d x+i)}{2 d^2}+\frac{x}{d}\right )\right )}{d}+b e x \cot ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 146, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}f}{2}}-{\frac{a{c}^{2}f}{2\,{d}^{2}}}+axe+{\frac{ace}{d}}+{\frac{b{\rm arccot} \left (dx+c\right )f{x}^{2}}{2}}-{\frac{{\rm arccot} \left (dx+c\right )b{c}^{2}f}{2\,{d}^{2}}}+{\rm arccot} \left (dx+c\right )xbe+{\frac{b{\rm arccot} \left (dx+c\right )ce}{d}}+{\frac{bfx}{2\,d}}+{\frac{cbf}{2\,{d}^{2}}}-{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) cf}{2\,{d}^{2}}}+{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) e}{2\,d}}-{\frac{bf\arctan \left ( dx+c \right ) }{2\,{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47205, size = 153, normalized size = 1.58 \begin{align*} \frac{1}{2} \, a f x^{2} + \frac{1}{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 f + a 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 )} b e}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.30109, size = 257, normalized size = 2.65 \begin{align*} \frac{a d^{2} f x^{2} +{\left (2 \, a d^{2} e + b d f\right )} x +{\left (b d^{2} f x^{2} + 2 \, b d^{2} e x\right )} \operatorname{arccot}\left (d x + c\right ) -{\left (2 \, b c d e -{\left (b c^{2} - b\right )} f\right )} \arctan \left (d x + c\right ) +{\left (b d e - b c f\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{2 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.02614, size = 177, normalized size = 1.82 \begin{align*} \begin{cases} a e x + \frac{a f x^{2}}{2} - \frac{b c^{2} f \operatorname{acot}{\left (c + d x \right )}}{2 d^{2}} + \frac{b c e \operatorname{acot}{\left (c + d x \right )}}{d} - \frac{b c f \log{\left (\frac{c^{2}}{d^{2}} + \frac{2 c x}{d} + x^{2} + \frac{1}{d^{2}} \right )}}{2 d^{2}} + b e x \operatorname{acot}{\left (c + d x \right )} + \frac{b f x^{2} \operatorname{acot}{\left (c + d x \right )}}{2} + \frac{b e \log{\left (\frac{c^{2}}{d^{2}} + \frac{2 c x}{d} + x^{2} + \frac{1}{d^{2}} \right )}}{2 d} + \frac{b f x}{2 d} + \frac{b f \operatorname{acot}{\left (c + d x \right )}}{2 d^{2}} & \text{for}\: d \neq 0 \\\left (a + b \operatorname{acot}{\left (c \right )}\right ) \left (e x + \frac{f x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23385, size = 340, normalized size = 3.51 \begin{align*} \frac{2 \, \pi b d^{2} f x^{2} \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^{2} f x^{2} \mathrm{sgn}\left (d x + c\right ) + \pi b d^{2} f x^{2} + 2 \, b d^{2} f x^{2} \arctan \left (\frac{1}{d x + c}\right ) + 2 \, a d^{2} f x^{2} + 4 \, b d^{2} x \arctan \left (\frac{1}{d x + c}\right ) e + \pi b c^{2} f \mathrm{sgn}\left (d x + c\right ) - \pi b c^{2} f - 2 \, b c^{2} f \arctan \left (\frac{1}{d x + c}\right ) + 4 \, a d^{2} x e - 4 \, b c d \arctan \left (d x + c\right ) e + 2 \, b d f x - 2 \, b c f \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 2 \, b d e \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - \pi b f \mathrm{sgn}\left (d x + c\right ) + \pi b f + 2 \, b f \arctan \left (\frac{1}{d x + c}\right )}{4 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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