\(\int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx\) [197]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [C] (warning: unable to verify)
   Fricas [B] (verification not implemented)
   Sympy [F(-2)]
   Maxima [A] (verification not implemented)
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 20, antiderivative size = 116 \[ \int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx=-\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 a+2 b x}\right )}{8 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 116, 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 = {5304, 2215, 2221, 2611, 2320, 6724} \[ \int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx=-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 a+2 b x}\right )}{8 b^2}+\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac {1}{6} i b x^3 \]

[In]

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

[Out]

(-1/6*I)*b*x^3 + (x^2*ArcCot[c - (I - c)*Tanh[a + b*x]])/2 + (I/4)*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] + ((I/4)*x
*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b - ((I/8)*PolyLog[3, I*c*E^(2*a + 2*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 5304

Int[ArcCot[(c_.) + (d_.)*Tanh[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*(ArcCot[c + d*Tanh[a + b*x]]/(f*(m + 1))), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d + c*E^(2
*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - 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-(i-c) \tanh (a+b x))+\frac {1}{2} b \int \frac {x^2}{i+c e^{2 a+2 b x}} \, dx \\ & = -\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x))+\frac {1}{2} (i b c) \int \frac {e^{2 a+2 b x} x^2}{i+c e^{2 a+2 b x}} \, dx \\ & = -\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )-\frac {1}{2} i \int x \log \left (1-i c e^{2 a+2 b x}\right ) \, dx \\ & = -\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \int \operatorname {PolyLog}\left (2,i c e^{2 a+2 b x}\right ) \, dx}{4 b} \\ & = -\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i c x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^2} \\ & = -\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 a+2 b x}\right )}{8 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89 \[ \int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx=\frac {2 b^2 x^2 \left (2 \cot ^{-1}(c+(-i+c) \tanh (a+b x))+i \log \left (1+\frac {i e^{-2 (a+b x)}}{c}\right )\right )-2 i b x \operatorname {PolyLog}\left (2,-\frac {i e^{-2 (a+b x)}}{c}\right )-i \operatorname {PolyLog}\left (3,-\frac {i e^{-2 (a+b x)}}{c}\right )}{8 b^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 2.03 (sec) , antiderivative size = 1373, normalized size of antiderivative = 11.84

method result size
risch \(\text {Expression too large to display}\) \(1373\)

[In]

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

[Out]

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

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. 246 vs. \(2 (83) = 166\).

Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.12 \[ \int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx=\frac {-2 i \, b^{3} x^{3} + 3 i \, b^{2} x^{2} \log \left (\frac {{\left (c - i\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c e^{\left (2 \, b x + 2 \, a\right )} + i}\right ) - 2 i \, a^{3} + 6 i \, b x {\rm Li}_2\left (\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (b x + a\right )}\right ) + 6 i \, b x {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (b x + a\right )}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (b x + a\right )} + i \, \sqrt {4 i \, c}}{2 \, c}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (b x + a\right )} - i \, \sqrt {4 i \, c}}{2 \, c}\right ) - 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (b x + a\right )} + 1\right ) - 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (-\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (b x + a\right )} + 1\right ) - 6 i \, {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {4 i \, c} e^{\left (b x + a\right )}\right ) - 6 i \, {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (b x + a\right )}\right )}{12 \, b^{2}} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 1.35 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx={\left (\frac {2 \, x^{3}}{3 i \, c + 3} - \frac {2 \, b^{2} x^{2} \log \left (-i \, c e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (i \, c e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(i \, c e^{\left (2 \, b x + 2 \, a\right )})}{-2 \, b^{3} {\left (-i \, c - 1\right )}}\right )} b {\left (c - i\right )} + \frac {1}{2} \, x^{2} \operatorname {arccot}\left ({\left (c - i\right )} \tanh \left (b x + a\right ) + c\right ) \]

[In]

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

[Out]

(2*x^3/(3*I*c + 3) - (2*b^2*x^2*log(-I*c*e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(I*c*e^(2*b*x + 2*a)) - polylog(3,
I*c*e^(2*b*x + 2*a)))/(b^3*(2*I*c + 2)))*b*(c - I) + 1/2*x^2*arccot((c - I)*tanh(b*x + a) + c)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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