\(\int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx\) [189]

Optimal result

Integrand size = 13, antiderivative size = 267 \[ \int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx=\frac {1}{2} x^2 \cot ^{-1}(c+d \tanh (a+b x))-\frac {1}{4} i x^2 \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i \operatorname {PolyLog}\left (3,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{8 b^2}-\frac {i \operatorname {PolyLog}\left (3,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{8 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5308, 2221, 2611, 2320, 6724} \[ \int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx=\frac {i \operatorname {PolyLog}\left (3,-\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )}{8 b^2}-\frac {i \operatorname {PolyLog}\left (3,-\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )}{8 b^2}-\frac {i x \operatorname {PolyLog}\left (2,-\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )}{4 b}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )}{4 b}-\frac {1}{4} i x^2 \log \left (1+\frac {(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {(c+d+i) e^{2 a+2 b x}}{c-d+i}\right )+\frac {1}{2} x^2 \cot ^{-1}(d \tanh (a+b x)+c) \]

[In]

Int[x*ArcCot[c + d*Tanh[a + b*x]],x]

[Out]

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

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 5308

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[I*b*((I - c - d)/(f*(m + 1))), Int[(e + f*x)^(m +
1)*(E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x))), x], x] + Dist[I*b*((I + c + d)/(f*(m + 1))), I
nt[(e + f*x)^(m + 1)*(E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x))), x], x]) /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[m, 0] && NeQ[(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+d \tanh (a+b x))-\frac {1}{2} (b (1-i (c+d))) \int \frac {e^{2 a+2 b x} x^2}{i+c-d+(i+c+d) e^{2 a+2 b x}} \, dx+\frac {1}{2} (b (1+i (c+d))) \int \frac {e^{2 a+2 b x} x^2}{i-c+d+(i-c-d) e^{2 a+2 b x}} \, dx \\ & = \frac {1}{2} x^2 \cot ^{-1}(c+d \tanh (a+b x))-\frac {1}{4} i x^2 \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )+\frac {1}{2} i \int x \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right ) \, dx-\frac {1}{2} i \int x \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right ) \, dx \\ & = \frac {1}{2} x^2 \cot ^{-1}(c+d \tanh (a+b x))-\frac {1}{4} i x^2 \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i \int \operatorname {PolyLog}\left (2,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right ) \, dx}{4 b}-\frac {i \int \operatorname {PolyLog}\left (2,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right ) \, dx}{4 b} \\ & = \frac {1}{2} x^2 \cot ^{-1}(c+d \tanh (a+b x))-\frac {1}{4} i x^2 \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {(-i+c+d) x}{-i+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^2}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {(i+c+d) x}{i+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^2} \\ & = \frac {1}{2} x^2 \cot ^{-1}(c+d \tanh (a+b x))-\frac {1}{4} i x^2 \log \left (1+\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )-\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{4 b}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{4 b}+\frac {i \operatorname {PolyLog}\left (3,-\frac {(i-c-d) e^{2 a+2 b x}}{i-c+d}\right )}{8 b^2}-\frac {i \operatorname {PolyLog}\left (3,-\frac {(i+c+d) e^{2 a+2 b x}}{i+c-d}\right )}{8 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.24 \[ \int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx=\frac {1}{2} x^2 \cot ^{-1}(c+d \tanh (a+b x))+\frac {d \left (2 b^2 x^2 \log \left (1+\frac {2 \left (1+(c+d)^2\right ) e^{2 (a+b x)}}{2+2 c^2-2 d^2-4 \sqrt {-d^2}}\right )-2 b^2 x^2 \log \left (1+\frac {\left (1+(c+d)^2\right ) e^{2 (a+b x)}}{1+c^2-d^2+2 \sqrt {-d^2}}\right )+2 b x \operatorname {PolyLog}\left (2,\frac {\left (1+c^2+2 c d+d^2\right ) e^{2 (a+b x)}}{-1-c^2+d^2+2 \sqrt {-d^2}}\right )-2 b x \operatorname {PolyLog}\left (2,-\frac {\left (1+(c+d)^2\right ) e^{2 (a+b x)}}{1+c^2-d^2+2 \sqrt {-d^2}}\right )-\operatorname {PolyLog}\left (3,-\frac {2 \left (1+(c+d)^2\right ) e^{2 (a+b x)}}{2+2 c^2-2 d^2-4 \sqrt {-d^2}}\right )+\operatorname {PolyLog}\left (3,-\frac {\left (1+(c+d)^2\right ) e^{2 (a+b x)}}{1+c^2-d^2+2 \sqrt {-d^2}}\right )\right )}{8 b^2 \sqrt {-d^2}} \]

[In]

Integrate[x*ArcCot[c + d*Tanh[a + b*x]],x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 4.51 (sec) , antiderivative size = 6566, normalized size of antiderivative = 24.59

method	result	size
risch	\(\text {Expression too large to display}\)	\(6566\)

[In]

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

[Out]

result too large to display

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. 1067 vs. \(2 (197) = 394\).

Time = 0.35 (sec) , antiderivative size = 1067, normalized size of antiderivative = 4.00 \[ \int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(2*b^2*x^2*arctan(cosh(b*x + a)/(c*cosh(b*x + a) + d*sinh(b*x + a))) - 2*I*b*x*dilog(sqrt(-(c^2 - d^2 + 2*
I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))) - 2*I*b*x*dilog(-sqrt(-(c^2 - d^2 + 2*I*d +
1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))) + 2*I*b*x*dilog(sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c
^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))) + 2*I*b*x*dilog(-sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 -
2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a))) - I*a^2*log(2*(c^2 + 2*c*d + d^2 + 1)*cosh(b*x + a) + 2*(c^
2 + 2*c*d + d^2 + 1)*sinh(b*x + a) + 2*(c^2 - d^2 - 2*I*d + 1)*sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^
2 + 1))) - I*a^2*log(2*(c^2 + 2*c*d + d^2 + 1)*cosh(b*x + a) + 2*(c^2 + 2*c*d + d^2 + 1)*sinh(b*x + a) - 2*(c^
2 - d^2 - 2*I*d + 1)*sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))) + I*a^2*log(2*(c^2 + 2*c*d + d^2
+ 1)*cosh(b*x + a) + 2*(c^2 + 2*c*d + d^2 + 1)*sinh(b*x + a) + 2*(c^2 - d^2 + 2*I*d + 1)*sqrt(-(c^2 - d^2 - 2*
I*d + 1)/(c^2 - 2*c*d + d^2 + 1))) + I*a^2*log(2*(c^2 + 2*c*d + d^2 + 1)*cosh(b*x + a) + 2*(c^2 + 2*c*d + d^2
+ 1)*sinh(b*x + a) - 2*(c^2 - d^2 + 2*I*d + 1)*sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))) + (-I*b
^2*x^2 + I*a^2)*log(sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1
) + (-I*b^2*x^2 + I*a^2)*log(-sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(b*x
+ a)) + 1) + (I*b^2*x^2 - I*a^2)*log(sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) +
sinh(b*x + a)) + 1) + (I*b^2*x^2 - I*a^2)*log(-sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*
x + a) + sinh(b*x + a)) + 1) + 2*I*polylog(3, sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x
+ a) + sinh(b*x + a))) + 2*I*polylog(3, -sqrt(-(c^2 - d^2 + 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a
) + sinh(b*x + a))) - 2*I*polylog(3, sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + s
inh(b*x + a))) - 2*I*polylog(3, -sqrt(-(c^2 - d^2 - 2*I*d + 1)/(c^2 - 2*c*d + d^2 + 1))*(cosh(b*x + a) + sinh(
b*x + a))))/b^2

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/2*x^2*arctan2(e^(2*b*x + 2*a) + 1, (c*e^(2*a) + d*e^(2*a))*e^(2*b*x) + c - d) + 2*b*d*integrate(x^2*e^(2*b*x
+ 2*a)/(c^2 - 2*c*d + d^2 + (c^2*e^(4*a) + 2*c*d*e^(4*a) + d^2*e^(4*a) + e^(4*a))*e^(4*b*x) + 2*(c^2*e^(2*a)
- d^2*e^(2*a) + e^(2*a))*e^(2*b*x) + 1), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*acot(c + d*tanh(a + b*x)),x)

[Out]

int(x*acot(c + d*tanh(a + b*x)), x)