3.166 \(\int x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx\)

Optimal. Leaf size=155 \[ \frac {\text {Li}_4\left (-i c e^{2 i a+2 i b x}\right )}{8 b^3}-\frac {i x \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac {x^2 \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{3} x^3 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {b x^4}{12} \]

[Out]

-1/12*b*x^4+1/3*x^3*arccot(c-(1-I*c)*tan(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

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Rubi [A]  time = 0.26, 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 = {5172, 2184, 2190, 2531, 6609, 2282, 6589} \[ -\frac {i x \text {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac {\text {PolyLog}\left (4,-i c e^{2 i a+2 i b x}\right )}{8 b^3}-\frac {x^2 \text {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{6} i x^3 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{3} x^3 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {b x^4}{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x^4)/12 + (x^3*ArcCot[c - (1 - I*c)*Tan[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 2184

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282

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 2531

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 5172

Int[ArcCot[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^(m
+ 1)*ArcCot[c + d*Tan[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 6589

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 6609

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) \tan (a+b x)) \, dx &=\frac {1}{3} x^3 \cot ^{-1}(c-(1-i c) \tan (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) \tan (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) \tan (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) \tan (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) \tan (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) \tan (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 {\operatorname {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) \tan (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*}

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Mathematica [A]  time = 0.26, size = 141, normalized size = 0.91 \[ \frac {1}{3} x^3 \cot ^{-1}(c+i (c+i) \tan (a+b x))-\frac {4 i b^3 x^3 \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )-6 b^2 x^2 \text {Li}_2\left (\frac {i e^{-2 i (a+b x)}}{c}\right )+6 i b x \text {Li}_3\left (\frac {i e^{-2 i (a+b x)}}{c}\right )+3 \text {Li}_4\left (\frac {i e^{-2 i (a+b x)}}{c}\right )}{24 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [C]  time = 0.72, size = 321, normalized size = 2.07 \[ -\frac {b^{4} x^{4} + 2 i \, b^{3} x^{3} \log \left (\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) - a^{4} - 2 i \, a^{3} \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - 2 i \, a^{3} \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + 12 i \, b x {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 12 i \, b x {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) - {\left (-2 i \, b^{3} x^{3} - 2 i \, a^{3}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) - {\left (-2 i \, b^{3} x^{3} - 2 i \, a^{3}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) - 12 \, {\rm polylog}\left (4, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) - 12 \, {\rm polylog}\left (4, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right )}{12 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(b^4*x^4 + 2*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(1/
2*sqrt(-4*I*c)*e^(I*b*x + I*a)) + 6*b^2*x^2*dilog(-1/2*sqrt(-4*I*c)*e^(I*b*x + I*a)) - a^4 - 2*I*a^3*log(1/2*(
2*c*e^(I*b*x + I*a) + I*sqrt(-4*I*c))/c) - 2*I*a^3*log(1/2*(2*c*e^(I*b*x + I*a) - I*sqrt(-4*I*c))/c) + 12*I*b*
x*polylog(3, 1/2*sqrt(-4*I*c)*e^(I*b*x + I*a)) + 12*I*b*x*polylog(3, -1/2*sqrt(-4*I*c)*e^(I*b*x + I*a)) - (-2*
I*b^3*x^3 - 2*I*a^3)*log(1/2*sqrt(-4*I*c)*e^(I*b*x + I*a) + 1) - (-2*I*b^3*x^3 - 2*I*a^3)*log(-1/2*sqrt(-4*I*c
)*e^(I*b*x + I*a) + 1) - 12*polylog(4, 1/2*sqrt(-4*I*c)*e^(I*b*x + I*a)) - 12*polylog(4, -1/2*sqrt(-4*I*c)*e^(
I*b*x + I*a)))/b^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arccot}\left (-{\left (-i \, c + 1\right )} \tan \left (b x + a\right ) + c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 6.51, size = 1527, normalized size = 9.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccot(c-(1-I*c)*tan(b*x+a)),x)

[Out]

-1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))+1))*csgn(I*(I+c))*csgn(I/(exp(2*I*(b*x+a))+1)*(I+c))-1/12*x^3*Pi*csgn(I*
exp(2*I*(b*x+a)))*csgn(I/(exp(2*I*(b*x+a))+1)*(I+c))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))-1/4*x
^2*polylog(2,-I*exp(2*I*(b*x+a))*c)/b+1/6*I*x^3*ln(exp(2*I*(b*x+a))*c-I)-1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))+
1)*(I+c))^3-1/12*b*x^4-1/2*I/b^2*a^2*ln(1-I*exp(I*(b*x+a))*(I*c)^(1/2))*x+1/2*I/b^2*ln(1+I*c*exp(2*I*(b*x+a)))
*x*a^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))*(I+
c)/(exp(2*I*(b*x+a))+1))*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))-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))*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))-1/6*I*x^3*ln(I+c)-1/3*I*x
^3*ln(exp(I*(b*x+a)))+1/4/b^3*polylog(2,-I*exp(2*I*(b*x+a))*c)*a^2+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I
*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))^2+1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))+1)*(I+c))*csgn(I*exp(2*I*
(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))^2+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))*csgn(ex
p(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))^2+1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))+1))*csgn(I/(exp(2*I*(b*x+a))
+1)*(I+c))^2-1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))^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(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))+1))^2-1/12*x^3*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/
(exp(2*I*(b*x+a))+1))^3+1/6*x^3*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2+1/12*x^3*Pi*csgn(I*(exp(2
*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))+1))^3-1/6*I*x^3*ln(1+I*c*exp(2*I*(b*x+a)))-1/2/b^3*a^2*dilog(1+I*exp(I*(b*x
+a))*(I*c)^(1/2))-1/2/b^3*a^2*dilog(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))+1/3*I/b^3*ln(1+I*c*exp(2*I*(b*x+a)))*a^3+1/12*x^3*Pi*csgn(I*(I+c))*csgn(I/(exp(2*I*(b*x+a))+1)*(I+c))^2
-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*cs
gn(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))^2-1/2*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))*(I+c)/(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/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))-
1/4*I*x*polylog(3,-I*exp(2*I*(b*x+a))*c)/b^2-1/12*x^3*Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))+1/1
2*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))

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maxima [B]  time = 0.37, size = 312, normalized size = 2.01 \[ -\frac {\frac {{\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \operatorname {arccot}\left ({\left (-i \, c + 1\right )} \tan \left (b x + a\right ) - c\right )}{b^{2}} + \frac {3 \, {\left (-3 i \, {\left (b x + a\right )}^{4} + 12 i \, {\left (b x + a\right )}^{3} a - 18 i \, {\left (b x + a\right )}^{2} a^{2} + {\left (8 i \, {\left (b x + a\right )}^{3} - 18 i \, {\left (b x + a\right )}^{2} a + 18 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), -c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + {\left (-12 i \, {\left (b x + a\right )}^{2} + 18 i \, {\left (b x + a\right )} a - 9 i \, a^{2}\right )} {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 9 \, {\left (b x + a\right )} a^{2}\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left (4 \, b x + a\right )} {\rm Li}_{3}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}) + 6 i \, {\rm Li}_{4}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )} {\left (i \, c - 1\right )}}{b^{2} {\left (12 \, c + 12 i\right )}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2)*arccot((-I*c + 1)*tan(b*x + a) - c)/b^2 + 3*(-3*I*(b*x
+ a)^4 + 12*I*(b*x + a)^3*a - 18*I*(b*x + a)^2*a^2 + (8*I*(b*x + a)^3 - 18*I*(b*x + a)^2*a + 18*I*(b*x + a)*a
^2)*arctan2(c*cos(2*b*x + 2*a), -c*sin(2*b*x + 2*a) + 1) + (-12*I*(b*x + a)^2 + 18*I*(b*x + a)*a - 9*I*a^2)*di
log(-I*c*e^(2*I*b*x + 2*I*a)) + (4*(b*x + a)^3 - 9*(b*x + a)^2*a + 9*(b*x + a)*a^2)*log(c^2*cos(2*b*x + 2*a)^2
+ c^2*sin(2*b*x + 2*a)^2 - 2*c*sin(2*b*x + 2*a) + 1) + 3*(4*b*x + a)*polylog(3, -I*c*e^(2*I*b*x + 2*I*a)) + 6
*I*polylog(4, -I*c*e^(2*I*b*x + 2*I*a)))*(I*c - 1)/(b^2*(12*c + 12*I)))/b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {acot}\left (c+\mathrm {tan}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: CoercionFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: CoercionFailed

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