\(\int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx\) [167]

Optimal result

Integrand size = 20, antiderivative size = 124 \[ \int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=-\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{8 b^2} \]

[Out]

-1/6*b*x^3+1/2*x^2*arccot(c-(1-I*c)*tan(b*x+a))-1/4*I*x^2*ln(1+I*c*exp(2*I*a+2*I*b*x))-1/4*x*polylog(2,-I*c*ex
p(2*I*a+2*I*b*x))/b-1/8*I*polylog(3,-I*c*exp(2*I*a+2*I*b*x))/b^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5280, 2215, 2221, 2611, 2320, 6724} \[ \int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=-\frac {i \operatorname {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{8 b^2}-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {b x^3}{6} \]

[In]

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

[Out]

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

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 5280

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))+\frac {1}{2} (i b) \int \frac {x^2}{i (-1+i c)+c+c e^{2 i a+2 i b x}} \, dx \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))+\frac {1}{2} (b c) \int \frac {e^{2 i a+2 i b x} x^2}{i (-1+i c)+c+c e^{2 i a+2 i b x}} \, dx \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {1}{2} i \int x \log \left (1+\frac {c e^{2 i a+2 i b x}}{i (-1+i c)+c}\right ) \, dx \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {\int \operatorname {PolyLog}\left (2,-\frac {c e^{2 i a+2 i b x}}{i (-1+i c)+c}\right ) \, dx}{4 b} \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2} \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{8 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=\frac {1}{2} x^2 \cot ^{-1}(c+i (i+c) \tan (a+b x))-\frac {i \left (2 b^2 x^2 \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )+2 i b x \operatorname {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{c}\right )+\operatorname {PolyLog}\left (3,\frac {i e^{-2 i (a+b x)}}{c}\right )\right )}{8 b^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 2.64 (sec) , antiderivative size = 1414, normalized size of antiderivative = 11.40

method	result	size
risch	\(\text {Expression too large to display}\)	\(1414\)

[In]

int(x*arccot(c-(1-I*c)*tan(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/2*I/b^2*a^2*ln(1+I*exp(I*(b*x+a))*(I*c)^(1/2))-1/8*(Pi*csgn(I/(exp(2*I*(b*x+a))+1))*csgn(I*(I+c))*csgn(I/(ex
p(2*I*(b*x+a))+1)*(I+c))-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))-Pi*csgn(I/(exp(2*I*(b*x+a))+1))*csgn(I/(exp(2*I*(b*x+a))+1)*(I+c))^2+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+Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I*
exp(2*I*(b*x+a)))-2*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2+Pi*csgn(I*exp(2*I*(b*x+a)))^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))-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+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))-Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp
(2*I*(b*x+a))+1))^2-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))-Pi*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))+1))^2-Pi*csgn(I*(I+c))*csgn(I/(exp(2*I*(b*x+a)
)+1)*(I+c))^2+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+Pi*csgn(
I/(exp(2*I*(b*x+a))+1)*(I+c))^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-Pi*csgn(I*(exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))+1))^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+Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*
(b*x+a))+1))^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))^2+Pi*csgn((exp(2*I*(b*x+a))*c-I)/(exp(2*I*(b*x+a))+1))^3+Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*
x+a))+1))^3+2*I*ln(I+c))*x^2-1/4*I/b^2*a^2*ln(-exp(2*I*(b*x+a))*c+I)+1/2*I/b^2*a^2*ln(1-I*exp(I*(b*x+a))*(I*c)
^(1/2))-1/4*I/b^2*ln(I*exp(2*I*(b*x+a))*c+1)*a^2-1/4*I*ln(I*exp(2*I*(b*x+a))*c+1)*x^2-1/4/b*polylog(2,-I*exp(2
*I*(b*x+a))*c)*x-1/4/b^2*polylog(2,-I*exp(2*I*(b*x+a))*c)*a-1/2*I*x^2*ln(exp(I*(b*x+a)))-1/8*I/b^2*polylog(3,-
I*exp(2*I*(b*x+a))*c)+1/4*I*x^2*ln(exp(2*I*(b*x+a))*c-I)+1/2*I/b*a*ln(1-I*exp(I*(b*x+a))*(I*c)^(1/2))*x+1/2*I/
b*a*ln(1+I*exp(I*(b*x+a))*(I*c)^(1/2))*x-1/2*I/b*ln(I*exp(2*I*(b*x+a))*c+1)*a*x+1/2/b^2*a*dilog(1+I*exp(I*(b*x
+a))*(I*c)^(1/2))+1/2/b^2*a*dilog(1-I*exp(I*(b*x+a))*(I*c)^(1/2))-1/6*b*x^3

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. 270 vs. \(2 (86) = 172\).

Time = 0.27 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.18 \[ \int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=-\frac {2 \, b^{3} x^{3} + 3 i \, b^{2} x^{2} \log \left (\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) + 2 \, a^{3} + 6 \, b x {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 6 \, b x {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + 3 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) + 3 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) + 6 i \, {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 6 i \, {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right )}{12 \, b^{2}} \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]

[In]

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

[Out]

Exception raised: CoercionFailed >> Cannot convert -_t0**4 - 3*_t0**2*I*c*exp(2*I*a) + _t0**2*exp(2*I*a) + 2*c
**2*exp(4*I*a) + I*c*exp(4*I*a) of type <class 'sympy.core.add.Add'> to QQ_I[x,b,c,_t0,exp(I*a)]

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. 220 vs. \(2 (86) = 172\).

Time = 0.20 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.77 \[ \int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=-\frac {\frac {6 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \operatorname {arccot}\left ({\left (-i \, c + 1\right )} \tan \left (b x + a\right ) - c\right )}{b} + \frac {{\left (-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 6 i \, b x {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (-i \, {\left (b x + a\right )}^{2} + 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), -c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\rm Li}_{3}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )} {\left (i \, c - 1\right )}}{b {\left (c + i\right )}}}{12 \, b} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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