\(\int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx\) [240]

Optimal result

Integrand size = 20, antiderivative size = 170 \[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {i x^2 \operatorname {PolyLog}\left (2,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (3,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (4,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{8 b^3} \]

[Out]

1/12*I*b*x^4+1/3*x^3*arccoth(1-I*d+d*tan(b*x+a))-1/6*x^3*ln(1+(1-I*d)*exp(2*I*a+2*I*b*x))+1/4*I*x^2*polylog(2,
-(1-I*d)*exp(2*I*a+2*I*b*x))/b-1/4*x*polylog(3,-(1-I*d)*exp(2*I*a+2*I*b*x))/b^2-1/8*I*polylog(4,-(1-I*d)*exp(2
*I*a+2*I*b*x))/b^3

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6399, 2215, 2221, 2611, 6744, 2320, 6724} \[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=-\frac {i \operatorname {PolyLog}\left (4,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{8 b^3}-\frac {x \operatorname {PolyLog}\left (3,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b^2}+\frac {i x^2 \operatorname {PolyLog}\left (2,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b}-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \tan (a+b x)-i d+1)+\frac {1}{12} i b x^4 \]

[In]

Int[x^2*ArcCoth[1 - I*d + d*Tan[a + b*x]],x]

[Out]

(I/12)*b*x^4 + (x^3*ArcCoth[1 - I*d + d*Tan[a + b*x]])/3 - (x^3*Log[1 + (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/6
+ ((I/4)*x^2*PolyLog[2, -((1 - I*d)*E^((2*I)*a + (2*I)*b*x))])/b - (x*PolyLog[3, -((1 - I*d)*E^((2*I)*a + (2*I
)*b*x))])/(4*b^2) - ((I/8)*PolyLog[4, -((1 - I*d)*E^((2*I)*a + (2*I)*b*x))])/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 6399

Int[ArcCoth[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*(ArcCoth[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 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*} \text {integral}& = \frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))+\frac {1}{3} (i b) \int \frac {x^3}{1+(1-i d) e^{2 i a+2 i b x}} \, dx \\ & = \frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{3} (b (i+d)) \int \frac {e^{2 i a+2 i b x} x^3}{1+(1-i d) e^{2 i a+2 i b x}} \, dx \\ & = \frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {1}{2} \int x^2 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right ) \, dx \\ & = \frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {i x^2 \operatorname {PolyLog}\left (2,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b}-\frac {i \int x \operatorname {PolyLog}\left (2,(-1+i d) e^{2 i a+2 i b x}\right ) \, dx}{2 b} \\ & = \frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {i x^2 \operatorname {PolyLog}\left (2,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (3,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b^2}+\frac {\int \operatorname {PolyLog}\left (3,(-1+i d) e^{2 i a+2 i b x}\right ) \, dx}{4 b^2} \\ & = \frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {i x^2 \operatorname {PolyLog}\left (2,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (3,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b^2}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i (i+d) x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^3} \\ & = \frac {1}{12} i b x^4+\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {1}{6} x^3 \log \left (1+(1-i d) e^{2 i a+2 i b x}\right )+\frac {i x^2 \operatorname {PolyLog}\left (2,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (3,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{4 b^2}-\frac {i \operatorname {PolyLog}\left (4,-\left ((1-i d) e^{2 i a+2 i b x}\right )\right )}{8 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.91 \[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\frac {1}{3} x^3 \coth ^{-1}(1-i d+d \tan (a+b x))-\frac {4 b^3 x^3 \log \left (1+\frac {i e^{-2 i (a+b x)}}{i+d}\right )+6 i b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {i e^{-2 i (a+b x)}}{i+d}\right )+6 b x \operatorname {PolyLog}\left (3,-\frac {i e^{-2 i (a+b x)}}{i+d}\right )-3 i \operatorname {PolyLog}\left (4,-\frac {i e^{-2 i (a+b x)}}{i+d}\right )}{24 b^3} \]

[In]

Integrate[x^2*ArcCoth[1 - I*d + d*Tan[a + b*x]],x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.31 (sec) , antiderivative size = 2273, normalized size of antiderivative = 13.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(2273\)

[In]

int(x^2*arccoth(1-I*d+d*tan(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/2/b^2*d/(I+d)*ln(1-I*(I+d)*exp(2*I*(b*x+a)))*a^2*x-1/2/b^2*d*a^2/(I+d)*ln(1-I*exp(I*(b*x+a))*(-I*(I+d))^(1/2
))*x-1/2/b^2*d*a^2/(I+d)*ln(1+I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))*x+1/4*I/b*d/(I+d)*polylog(2,I*(I+d)*exp(2*I*(
b*x+a)))*x^2+1/2*I/b^3*d*a^2/(I+d)*dilog(1-I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))+1/2*I/b^3*d*a^2/(I+d)*dilog(1+I*
exp(I*(b*x+a))*(-I*(I+d))^(1/2))-1/2*I/b^2*a^2/(I+d)*ln(1-I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))*x-1/2*I/b^2*a^2/(
I+d)*ln(1+I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))*x+1/2*I/b^2/(I+d)*ln(1-I*(I+d)*exp(2*I*(b*x+a)))*a^2*x-1/4*I/b^3*
a^2*d/(I+d)*polylog(2,I*(I+d)*exp(2*I*(b*x+a)))+1/12*I*b*x^4-1/6*d/(I+d)*ln(1-I*(I+d)*exp(2*I*(b*x+a)))*x^3-1/
2/b^3*a^2/(I+d)*dilog(1-I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))-1/4/b/(I+d)*polylog(2,I*(I+d)*exp(2*I*(b*x+a)))*x^2
+1/4/b^3/(I+d)*polylog(2,I*(I+d)*exp(2*I*(b*x+a)))*a^2-1/2/b^3*a^2/(I+d)*dilog(1+I*exp(I*(b*x+a))*(-I*(I+d))^(
1/2))-1/6*I/(I+d)*ln(1-I*(I+d)*exp(2*I*(b*x+a)))*x^3-1/3*x^3*ln(exp(I*(b*x+a)))+1/3/b^3*d/(I+d)*ln(1-I*(I+d)*e
xp(2*I*(b*x+a)))*a^3-1/4/b^2*d/(I+d)*polylog(3,I*(I+d)*exp(2*I*(b*x+a)))*x-1/2/b^3*d*a^3/(I+d)*ln(1-I*exp(I*(b
*x+a))*(-I*(I+d))^(1/2))-1/2/b^3*d*a^3/(I+d)*ln(1+I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))+1/6/b^3*a^3*d/(I+d)*ln(I*
exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)+1/3*I/b^3/(I+d)*ln(1-I*(I+d)*exp(2*I*(b*x+a)))*a^3-1/4*I/b^2/(I+d)*poly
log(3,I*(I+d)*exp(2*I*(b*x+a)))*x-1/8*I/b^3*d/(I+d)*polylog(4,I*(I+d)*exp(2*I*(b*x+a)))+1/6*I/b^3*a^3/(I+d)*ln
(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)-1/2*I/b^3*a^3/(I+d)*ln(1-I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))-1/2*I/b^
3*a^3/(I+d)*ln(1+I*exp(I*(b*x+a))*(-I*(I+d))^(1/2))+1/8/b^3/(I+d)*polylog(4,I*(I+d)*exp(2*I*(b*x+a)))-1/12*(I*
Pi*csgn(I*d)*csgn(I*d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))^2+2*I*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b
*x+a)))^2-I*Pi*csgn(I*d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))^3-I*Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+
a))*d+I)/(exp(2*I*(b*x+a))+1))*csgn((I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))^2-I*Pi*csg
n(I*d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))*csgn(d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))-I*Pi*csgn(I/(exp(2*
I*(b*x+a))+1))*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))^2+I*Pi*csgn(I*d/(exp(2*I
*(b*x+a))+1)*exp(2*I*(b*x+a)))*csgn(d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))^2+I*Pi*csgn(d/(exp(2*I*(b*x+a))+1
)*exp(2*I*(b*x+a)))^2+I*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))+1))^2+I*Pi*csgn(
I/(exp(2*I*(b*x+a))+1))*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))+1))^2-I*Pi*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*
(b*x+a))+1))^3+I*Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))*csgn((I*exp(2*I*(b*
x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))-I*Pi*csgn(I/(exp(2*I*(b*x+a))+1))*csgn(I*exp(2*I*(b*x+a)))*c
sgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))+1))+I*Pi*csgn(I/(exp(2*I*(b*x+a))+1))*csgn(I*(I*exp(2*I*(b*x+a))+exp(
2*I*(b*x+a))*d+I))*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))-I*Pi*csgn(I*exp(2*I*
(b*x+a)))^3+I*Pi*csgn((I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))^3-I*Pi*csgn(I*exp(I*(b*x
+a)))^2*csgn(I*exp(2*I*(b*x+a)))-I*Pi*csgn((I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I*(b*x+a))+1))^2-I
*Pi*csgn(I*d)*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))+1))*csgn(I*d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))-I*
Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I))*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)/(exp(2*I
*(b*x+a))+1))^2+I*Pi*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))+1))*csgn(I*d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+
a)))^2-I*Pi*csgn(d/(exp(2*I*(b*x+a))+1)*exp(2*I*(b*x+a)))^3+I*Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d
+I)/(exp(2*I*(b*x+a))+1))^3+2*ln(d))*x^3+1/6*x^3*ln(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d+I)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (118) = 236\).

Time = 0.28 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.02 \[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\frac {i \, b^{4} x^{4} + 2 \, b^{3} x^{3} \log \left (\frac {{\left ({\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{d}\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )}\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )}\right ) - i \, a^{4} + 2 \, a^{3} \log \left (\frac {2 \, {\left (d + i\right )} e^{\left (i \, b x + i \, a\right )} + i \, \sqrt {4 i \, d - 4}}{2 \, {\left (d + i\right )}}\right ) + 2 \, a^{3} \log \left (\frac {2 \, {\left (d + i\right )} e^{\left (i \, b x + i \, a\right )} - i \, \sqrt {4 i \, d - 4}}{2 \, {\left (d + i\right )}}\right ) - 12 \, b x {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )}\right ) - 12 \, b x {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )}\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (\frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )} + 1\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )} + 1\right ) - 12 i \, {\rm polylog}\left (4, \frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )}\right ) - 12 i \, {\rm polylog}\left (4, -\frac {1}{2} \, \sqrt {4 i \, d - 4} e^{\left (i \, b x + i \, a\right )}\right )}{12 \, b^{3}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\int x^{2} \operatorname {acoth}{\left (d \tan {\left (a + b x \right )} - i d + 1 \right )}\, dx \]

[In]

integrate(x**2*acoth(1-I*d+d*tan(b*x+a)),x)

[Out]

Integral(x**2*acoth(d*tan(a + b*x) - I*d + 1), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (118) = 236\).

Time = 0.24 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.02 \[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\frac {\frac {12 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \operatorname {arcoth}\left (d \tan \left (b x + a\right ) - i \, d + 1\right )}{b^{2}} - \frac {-3 i \, {\left (b x + a\right )}^{4} + 12 i \, {\left (b x + a\right )}^{3} a - 18 i \, {\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (-4 i \, {\left (b x + a\right )}^{3} + 9 i \, {\left (b x + a\right )}^{2} a - 9 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (-d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), d \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 3 \, {\left (4 i \, {\left (b x + a\right )}^{2} - 6 i \, {\left (b x + a\right )} a + 3 i \, a^{2}\right )} {\rm Li}_2\left ({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 9 \, {\left (b x + a\right )} a^{2}\right )} \log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, d \sin \left (2 \, b x + 2 \, a\right ) + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left (4 \, b x + a\right )} {\rm Li}_{3}({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}) + 6 i \, {\rm Li}_{4}({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )})}{b^{2}}}{36 \, b} \]

[In]

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

[Out]

1/36*(12*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2)*arccoth(d*tan(b*x + a) - I*d + 1)/b^2 - (-3*I*(b*x
+ a)^4 + 12*I*(b*x + a)^3*a - 18*I*(b*x + a)^2*a^2 - 2*(-4*I*(b*x + a)^3 + 9*I*(b*x + a)^2*a - 9*I*(b*x + a)*a
^2)*arctan2(-d*cos(2*b*x + 2*a) + sin(2*b*x + 2*a), d*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) + 1) - 3*(4*I*(b*x +
a)^2 - 6*I*(b*x + a)*a + 3*I*a^2)*dilog((I*d - 1)*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((d^2 + 1)*cos(2*b*x + 2*a)^2 + (d^2 + 1)*sin(2*b*x + 2*a)^2 + 2*d*sin(2*b*x + 2*a) + 2*cos
(2*b*x + 2*a) + 1) + 3*(4*b*x + a)*polylog(3, (I*d - 1)*e^(2*I*b*x + 2*I*a)) + 6*I*polylog(4, (I*d - 1)*e^(2*I
*b*x + 2*I*a)))/b^2)/b

Giac [F]

\[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\int { x^{2} \operatorname {arcoth}\left (d \tan \left (b x + a\right ) - i \, d + 1\right ) \,d x } \]

[In]

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

[Out]

integrate(x^2*arccoth(d*tan(b*x + a) - I*d + 1), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx=\int x^2\,\mathrm {acoth}\left (d\,\mathrm {tan}\left (a+b\,x\right )+1-d\,1{}\mathrm {i}\right ) \,d x \]

[In]

int(x^2*acoth(d*tan(a + b*x) - d*1i + 1),x)

[Out]

int(x^2*acoth(d*tan(a + b*x) - d*1i + 1), x)