3.2.79 \(\int x^2 \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx\) [179]

Optimal. Leaf size=155 \[ \frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x^2 \text {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i x \text {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {\text {PolyLog}\left (4,-i c e^{2 i a+2 i b x}\right )}{8 b^3} \]

[Out]

1/12*b*x^4+1/3*x^3*(Pi-arccot(-c+(1+I*c)*cot(b*x+a)))+1/6*I*x^3*ln(1+I*c*exp(2*I*a+2*I*b*x))+1/4*x^2*polylog(2
,-I*c*exp(2*I*a+2*I*b*x))/b+1/4*I*x*polylog(3,-I*c*exp(2*I*a+2*I*b*x))/b^2-1/8*polylog(4,-I*c*exp(2*I*a+2*I*b*
x))/b^3

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5282, 2215, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {\text {Li}_4\left (-i c e^{2 i a+2 i b x}\right )}{8 b^3}+\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {b x^4}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]

[Out]

(b*x^4)/12 + (x^3*ArcCot[c - (1 + I*c)*Cot[a + b*x]])/3 + (I/6)*x^3*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] + (x^
2*PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/4)*x*PolyLog[3, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/b^2
- PolyLog[4, (-I)*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 5282

Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m +
1)*(ArcCot[c + d*Cot[a + b*x]]/(f*(m + 1))), x] + Dist[I*(b/(f*(m + 1))), Int[(e + f*x)^(m + 1)/(c - I*d - c*
E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int x^2 \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx &=\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{3} (i b) \int \frac {x^3}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{3} (b c) \int \frac {e^{2 i a+2 i b x} x^3}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {1}{2} i \int x^2 \log \left (1-\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {\int x \text {Li}_2\left (\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx}{2 b}\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {i \int \text {Li}_3\left (\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, dx}{4 b^2}\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_3(-i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^3}\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {\text {Li}_4\left (-i c e^{2 i a+2 i b x}\right )}{8 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.17, size = 136, normalized size = 0.88 \begin {gather*} \frac {1}{24} \left (8 x^3 \cot ^{-1}(c+(-1-i c) \cot (a+b x))+4 i x^3 \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )-\frac {6 x^2 \text {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{c}\right )}{b}+\frac {6 i x \text {PolyLog}\left (3,\frac {i e^{-2 i (a+b x)}}{c}\right )}{b^2}+\frac {3 \text {PolyLog}\left (4,\frac {i e^{-2 i (a+b x)}}{c}\right )}{b^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]

[Out]

(8*x^3*ArcCot[c + (-1 - I*c)*Cot[a + b*x]] + (4*I)*x^3*Log[1 - I/(c*E^((2*I)*(a + b*x)))] - (6*x^2*PolyLog[2,
I/(c*E^((2*I)*(a + b*x)))])/b + ((6*I)*x*PolyLog[3, I/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, I/(c*E^((2
*I)*(a + b*x)))])/b^3)/24

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.04, size = 1527, normalized size = 9.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(1527\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(Pi-arccot(-c+(1+I*c)*cot(b*x+a))),x,method=_RETURNVERBOSE)

[Out]

1/12*x^3*Pi*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-
1))^2-1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*c
sgn(I*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/6*I*x^3*ln(exp(2*I*(
b*x+a))*c-I)+1/4*x^2*polylog(2,-I*exp(2*I*(b*x+a))*c)/b-1/4/b^3*polylog(2,-I*exp(2*I*(b*x+a))*c)*a^2-1/3*I/b^3
*ln(1+I*c*exp(2*I*(b*x+a)))*a^3-1/6*I/b^3*a^3*ln(-exp(2*I*(b*x+a))*c+I)+1/2*I/b^3*a^3*ln(1+I*exp(I*(b*x+a))*(I
*c)^(1/2))+1/2*I/b^3*a^3*ln(1-I*exp(I*(b*x+a))*(I*c)^(1/2))+1/12*x^3*Pi*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))^3+1
/2/b^3*a^2*dilog(1+I*exp(I*(b*x+a))*(I*c)^(1/2))+1/12*b*x^4+1/3*I*x^3*ln(exp(I*(b*x+a)))-1/12*x^3*Pi*csgn(I*(e
xp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))^3+1/12*Pi*x^3*csgn(I*exp(2*I*(b*x+a)))^3-1/6*Pi*x^3*csgn(I*exp(I*(b
*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2+1/12*Pi*x^3*csgn(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))+1/2/b^3*a^2*d
ilog(1-I*exp(I*(b*x+a))*(I*c)^(1/2))-1/2*I/b^2*ln(1+I*c*exp(2*I*(b*x+a)))*x*a^2+1/2*I/b^2*a^2*ln(1+I*exp(I*(b*
x+a))*(I*c)^(1/2))*x+1/2*I/b^2*a^2*ln(1-I*exp(I*(b*x+a))*(I*c)^(1/2))*x-1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1
))*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*csgn(I*(c-I))*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))^2+1/12*x^
3*Pi*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^3-1/8*polylog(4,-I*exp(2*I*(b*x+a))*c)/b^3+1/6*I*x^3*
ln(1+I*c*exp(2*I*(b*x+a)))+1/4*I*x*polylog(3,-I*exp(2*I*(b*x+a))*c)/b^2-1/12*x^3*Pi*csgn((exp(2*I*(b*x+a))*c-I
)/(exp(2*I*(b*x+a))-1))^3+1/12*x^3*Pi*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*csgn(I*e
xp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*cs
gn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))-1))+1/12*x^3*P
i*csgn(I*exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))-1/12*x
^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*c-I))*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+
a))-1))+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*(c-I)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(c-I)/
(exp(2*I*(b*x+a))-1))+1/12*x^3*Pi*csgn(exp(2*I*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*csgn(exp(2*I
*(b*x+a))*(c-I)/(exp(2*I*(b*x+a))-1))^3+1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(c-I))*csgn(I*(c-I)/(e
xp(2*I*(b*x+a))-1))+1/6*I*x^3*ln(c-I)+1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*c-I)/(
exp(2*I*(b*x+a))-1))^2+1/12*x^3*Pi*csgn(I*(exp(2*I*(b*x+a))*c-I))*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+
a))-1))^2

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(pi-arccot(-c+(1+I*c)*cot(b*x+a))),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for mor
e details)Is

________________________________________________________________________________________

Fricas [A]
time = 2.67, size = 174, normalized size = 1.12 \begin {gather*} \frac {2 \, b^{4} x^{4} + 8 \, \pi b^{3} x^{3} + 4 i \, b^{3} x^{3} \log \left (\frac {{\left (c - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 2 \, a^{4} - 4 i \, a^{3} \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + 6 i \, b x {\rm polylog}\left (3, -i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 4 \, {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \, {\rm polylog}\left (4, -i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(pi-arccot(-c+(1+I*c)*cot(b*x+a))),x, algorithm="fricas")

[Out]

1/24*(2*b^4*x^4 + 8*pi*b^3*x^3 + 4*I*b^3*x^3*log((c - I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I*a) - I)) + 6*
b^2*x^2*dilog(-I*c*e^(2*I*b*x + 2*I*a)) - 2*a^4 - 4*I*a^3*log((c*e^(2*I*b*x + 2*I*a) - I)/c) + 6*I*b*x*polylog
(3, -I*c*e^(2*I*b*x + 2*I*a)) - 4*(-I*b^3*x^3 - I*a^3)*log(I*c*e^(2*I*b*x + 2*I*a) + 1) - 3*polylog(4, -I*c*e^
(2*I*b*x + 2*I*a)))/b^3

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(pi-acot(-c+(1+I*c)*cot(b*x+a))),x)

[Out]

Exception raised: CoercionFailed >> Cannot convert -_t0**2*I + 2*c*exp(2*I*a) - I*exp(2*I*a) of type <class 's
ympy.core.add.Add'> to QQ_I[x,b,c,_t0,exp(I*a)]

________________________________________________________________________________________

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(x^2*(pi-arccot(-c+(1+I*c)*cot(b*x+a))),x, algorithm="giac")

[Out]

integrate((pi - arccot((I*c + 1)*cot(b*x + a) - c))*x^2, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\left (\Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(Pi + acot(c - cot(a + b*x)*(c*1i + 1))),x)

[Out]

int(x^2*(Pi + acot(c - cot(a + b*x)*(c*1i + 1))), x)

________________________________________________________________________________________