3.2.43 \(\int (a+b \cot ^{-1}(c+d x))^3 \, dx\) [143]

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

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 143, normalized size of antiderivative = 1.00, 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 = {5148, 4931, 5041, 4965, 5005, 5115, 6745} \begin {gather*} \frac {3 i b^2 \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{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}-\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d} \end {gather*}

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 4931

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

Rule 4965

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(
Log[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 5005

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 5041

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 5115

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cCot[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 5148

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 6745

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 {\text {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) \text {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) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.23, size = 228, normalized size = 1.59 \begin {gather*} \frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \cot ^{-1}(c+d x)+3 a^2 b \log \left (1+(c+d x)^2\right )+6 a b^2 \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \text {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )+2 b^3 \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \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 )-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 )\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (136 ) = 272\).
time = 1.10, size = 430, normalized size = 3.01

method	result	size
derivativedivides	\(\frac {\left (d x +c \right ) a^{3}+3 i \mathrm {arccot}\left (d x +c \right )^{2} a \,b^{2}+\mathrm {arccot}\left (d x +c \right )^{3} b^{3} \left (d x +c \right )-3 \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right )^{2} b^{3}-3 \mathrm {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+i \mathrm {arccot}\left (d x +c \right )^{3} b^{3}+6 i \polylog \left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}-6 \polylog \left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}-6 \polylog \left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+6 i \polylog \left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}+3 \mathrm {arccot}\left (d x +c \right )^{2} a \,b^{2} \left (d x +c \right )-6 \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right ) a \,b^{2}-6 \,\mathrm {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}+6 i \mathrm {arccot}\left (d x +c \right ) \polylog \left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+6 i \mathrm {arccot}\left (d x +c \right ) \polylog \left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+3 a^{2} b \,\mathrm {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {3 a^{2} b \ln \left (1+\left (d x +c \right )^{2}\right )}{2}}{d}\)	\(430\)
default	\(\frac {\left (d x +c \right ) a^{3}+3 i \mathrm {arccot}\left (d x +c \right )^{2} a \,b^{2}+\mathrm {arccot}\left (d x +c \right )^{3} b^{3} \left (d x +c \right )-3 \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right )^{2} b^{3}-3 \mathrm {arccot}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+i \mathrm {arccot}\left (d x +c \right )^{3} b^{3}+6 i \polylog \left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}-6 \polylog \left (3, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}-6 \polylog \left (3, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+6 i \polylog \left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}+3 \mathrm {arccot}\left (d x +c \right )^{2} a \,b^{2} \left (d x +c \right )-6 \ln \left (1-\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) \mathrm {arccot}\left (d x +c \right ) a \,b^{2}-6 \,\mathrm {arccot}\left (d x +c \right ) \ln \left (1+\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) a \,b^{2}+6 i \mathrm {arccot}\left (d x +c \right ) \polylog \left (2, \frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+6 i \mathrm {arccot}\left (d x +c \right ) \polylog \left (2, -\frac {d x +c +i}{\sqrt {1+\left (d x +c \right )^{2}}}\right ) b^{3}+3 a^{2} b \,\mathrm {arccot}\left (d x +c \right ) \left (d x +c \right )+\frac {3 a^{2} b \ln \left (1+\left (d x +c \right )^{2}\right )}{2}}{d}\)	\(430\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acot(c + d*x))^3,x)

[Out]

int((a + b*acot(c + d*x))^3, x)

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