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

Optimal. Leaf size=143 \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d}-\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d}+\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d} \]

[Out]

(I*(a + b*ArcCot[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcCot[c + d*x])^3)/d - (3*b*(a + b*ArcCot[c + d*x])^2*Log
[2/(1 + I*(c + d*x))])/d + ((3*I)*b^2*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d - (3*b^3*
PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d)

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Rubi [A]  time = 0.216905, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5040, 4847, 4921, 4855, 4885, 4995, 6610} \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d}-\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d}+\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCot[c + d*x])^3,x]

[Out]

(I*(a + b*ArcCot[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcCot[c + d*x])^3)/d - (3*b*(a + b*ArcCot[c + d*x])^2*Log
[2/(1 + I*(c + d*x))])/d + ((3*I)*b^2*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d - (3*b^3*
PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d)

Rule 5040

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

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 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 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 4995

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Cot[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \cot ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right )^2}{i-x} \, dx,x,c+d x\right )}{d}\\ &=\frac{i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1+i (c+d x)}\right )}{d}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1+i (c+d x)}\right )}{d}+\frac{3 i b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{d}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{i \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}+\frac{(c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1+i (c+d x)}\right )}{d}+\frac{3 i b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{d}-\frac{3 b^3 \text{Li}_3\left (1-\frac{2}{1+i (c+d x)}\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.312413, size = 228, normalized size = 1.59 \[ \frac{6 a b^2 \left (i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )+2 b^3 \left (-3 i \cot ^{-1}(c+d x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )+\frac{i \pi ^3}{8}\right )+3 a^2 b \log \left ((c+d x)^2+1\right )+6 a^2 b (c+d x) \cot ^{-1}(c+d x)+2 a^3 (c+d x)}{2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a^3*(c + d*x) + 6*a^2*b*(c + d*x)*ArcCot[c + d*x] + 3*a^2*b*Log[1 + (c + d*x)^2] + 6*a*b^2*(ArcCot[c + d*x]
*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x
])]) + 2*b^3*((I/8)*Pi^3 - I*ArcCot[c + d*x]^3 + (c + d*x)*ArcCot[c + d*x]^3 - 3*ArcCot[c + d*x]^2*Log[1 - E^(
(-2*I)*ArcCot[c + d*x])] - (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2
*I)*ArcCot[c + d*x])])/2))/(2*d)

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Maple [B]  time = 0.324, size = 507, normalized size = 3.6 \begin{align*} x{a}^{3}+{\frac{{a}^{3}c}{d}}+{\frac{3\,i \left ({\rm arccot} \left (dx+c\right ) \right ) ^{2}a{b}^{2}}{d}}+ \left ({\rm arccot} \left (dx+c\right ) \right ) ^{3}x{b}^{3}+{\frac{ \left ({\rm arccot} \left (dx+c\right ) \right ) ^{3}{b}^{3}c}{d}}-3\,{\frac{ \left ({\rm arccot} \left (dx+c\right ) \right ) ^{2}{b}^{3}}{d}\ln \left ( 1-{\frac{dx+c+i}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }-3\,{\frac{ \left ({\rm arccot} \left (dx+c\right ) \right ) ^{2}{b}^{3}}{d}\ln \left ( 1+{\frac{dx+c+i}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }+{\frac{6\,i{\rm arccot} \left (dx+c\right ){b}^{3}}{d}{\it polylog} \left ( 2,-{(dx+c+i){\frac{1}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}}} \right ) }+{\frac{6\,i{\rm arccot} \left (dx+c\right ){b}^{3}}{d}{\it polylog} \left ( 2,{(dx+c+i){\frac{1}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}}} \right ) }-6\,{\frac{{b}^{3}}{d}{\it polylog} \left ( 3,-{\frac{dx+c+i}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }-6\,{\frac{{b}^{3}}{d}{\it polylog} \left ( 3,{\frac{dx+c+i}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }+{\frac{6\,ia{b}^{2}}{d}{\it polylog} \left ( 2,{(dx+c+i){\frac{1}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}}} \right ) }+3\, \left ({\rm arccot} \left (dx+c\right ) \right ) ^{2}xa{b}^{2}+3\,{\frac{ \left ({\rm arccot} \left (dx+c\right ) \right ) ^{2}a{b}^{2}c}{d}}-6\,{\frac{{\rm arccot} \left (dx+c\right )a{b}^{2}}{d}\ln \left ( 1-{\frac{dx+c+i}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }-6\,{\frac{{\rm arccot} \left (dx+c\right )a{b}^{2}}{d}\ln \left ( 1+{\frac{dx+c+i}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }+{\frac{i \left ({\rm arccot} \left (dx+c\right ) \right ) ^{3}{b}^{3}}{d}}+{\frac{6\,ia{b}^{2}}{d}{\it polylog} \left ( 2,-{(dx+c+i){\frac{1}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}}} \right ) }+3\,{\rm arccot} \left (dx+c\right )x{a}^{2}b+3\,{\frac{{\rm arccot} \left (dx+c\right ){a}^{2}bc}{d}}+{\frac{3\,{a}^{2}b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*a^3+1/d*a^3*c+3*I/d*arccot(d*x+c)^2*a*b^2+arccot(d*x+c)^3*x*b^3+1/d*arccot(d*x+c)^3*b^3*c-3/d*arccot(d*x+c)^
2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*b^3-3/d*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*b^3+6*I/d*ar
ccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*b^3+6*I/d*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2
)^(1/2))*b^3-6/d*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*b^3-6/d*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*b^
3+6*I/d*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*a*b^2+3*arccot(d*x+c)^2*x*a*b^2+3/d*arccot(d*x+c)^2*a*b^2*c-6
/d*arccot(d*x+c)*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*a*b^2-6/d*arccot(d*x+c)*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2
))*a*b^2+I/d*arccot(d*x+c)^3*b^3+6*I/d*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))*a*b^2+3*arccot(d*x+c)*x*a^2*b
+3/d*arccot(d*x+c)*a^2*b*c+3/2/d*a^2*b*ln(1+(d*x+c)^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, b^{3} x \arctan \left (1, d x + c\right )^{3} - \frac{3}{32} \, b^{3} x \arctan \left (1, d x + c\right ) \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + a^{3} x + \frac{3 \,{\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^{2} b}{2 \, d} + \int \frac{28 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 4 \,{\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} d^{2} x^{2} + 96 \, a b^{2} \arctan \left (1, d x + c\right )^{2} + 4 \,{\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c^{2} + 4 \,{\left (3 \, b^{3} \arctan \left (1, d x + c\right )^{2} + 2 \,{\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c\right )} d x + 3 \,{\left (b^{3} d^{2} x^{2} \arctan \left (1, d x + c\right ) + b^{3} c^{2} \arctan \left (1, d x + c\right ) + b^{3} \arctan \left (1, d x + c\right ) +{\left (2 \, b^{3} c \arctan \left (1, d x + c\right ) - b^{3}\right )} d x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 12 \,{\left (b^{3} d^{2} x^{2} \arctan \left (1, d x + c\right ) + b^{3} c d x \arctan \left (1, d x + c\right )\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{32 \,{\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((a+b*arccot(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*b^3*x*arctan2(1, d*x + c)^3 - 3/32*b^3*x*arctan2(1, d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + a^3*x +
3/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a^2*b/d + integrate(1/32*(28*b^3*arctan2(1, d*x + c)^
3 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*x^2 + 96*a*b^2*arctan2(1, d*x + c)^2
+ 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2 + 4*(3*b^3*arctan2(1, d*x + c)^2 + 2*(7
*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*x + 3*(b^3*d^2*x^2*arctan2(1, d*x + c) + b^3
*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c) + (2*b^3*c*arctan2(1, d*x + c) - b^3)*d*x)*log(d^2*x^2 + 2*
c*d*x + c^2 + 1)^2 + 12*(b^3*d^2*x^2*arctan2(1, d*x + c) + b^3*c*d*x*arctan2(1, d*x + c))*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 (b^{3} \operatorname{arccot}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arccot}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arccot}\left (d x + c\right ) + a^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*arccot(d*x + c)^3 + 3*a*b^2*arccot(d*x + c)^2 + 3*a^2*b*arccot(d*x + c) + a^3, 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 )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acot(c + d*x))**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccot(d*x + c) + a)^3, x)