3.2.3 \(\int x \text {ArcTan}(c+(i+c) \coth (a+b x)) \, dx\) [103]

Optimal. Leaf size=113 \[ -\frac {1}{6} i b x^3+\frac {1}{2} x^2 \text {ArcTan}(c+(i+c) \coth (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \text {PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \text {PolyLog}\left (3,i c e^{2 a+2 b x}\right )}{8 b^2} \]

[Out]

-1/6*I*b*x^3+1/2*x^2*arctan(c+(I+c)*coth(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

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Rubi [A]
time = 0.14, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5305, 2215, 2221, 2611, 2320, 6724} \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c+(c+i) \coth (a+b x))-\frac {i \text {Li}_3\left (i c e^{2 a+2 b x}\right )}{8 b^2}+\frac {i x \text {Li}_2\left (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}{6} i b x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcTan[c + (I + c)*Coth[a + b*x]],x]

[Out]

(-1/6*I)*b*x^3 + (x^2*ArcTan[c + (I + c)*Coth[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 5305

Int[ArcTan[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*(ArcTan[c + d*Coth[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*} \int x \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(c+(i+c) \coth (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 \tan ^{-1}(c+(i+c) \coth (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 \tan ^{-1}(c+(i+c) \coth (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 \tan ^{-1}(c+(i+c) \coth (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \text {Li}_2\left (i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \int \text {Li}_2\left (i c e^{2 a+2 b x}\right ) \, dx}{4 b}\\ &=-\frac {1}{6} i b x^3+\frac {1}{2} x^2 \tan ^{-1}(c+(i+c) \coth (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \text {Li}_2\left (i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_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 \tan ^{-1}(c+(i+c) \coth (a+b x))+\frac {1}{4} i x^2 \log \left (1-i c e^{2 a+2 b x}\right )+\frac {i x \text {Li}_2\left (i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \text {Li}_3\left (i c e^{2 a+2 b x}\right )}{8 b^2}\\ \end {align*}

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Mathematica [A]
time = 1.17, size = 102, normalized size = 0.90 \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c+(i+c) \coth (a+b x))+\frac {i \left (2 b^2 x^2 \log \left (1+\frac {i e^{-2 (a+b x)}}{c}\right )-2 b x \text {PolyLog}\left (2,-\frac {i e^{-2 (a+b x)}}{c}\right )-\text {PolyLog}\left (3,-\frac {i e^{-2 (a+b x)}}{c}\right )\right )}{8 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcTan[c + (I + c)*Coth[a + b*x]],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 1442, normalized size = 12.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1442\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctan(c+(I+c)*coth(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.20, size = 106, normalized size = 0.94 \begin {gather*} {\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} \arctan \left ({\left (c + i\right )} \coth \left (b x + a\right ) + c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctan(c+(I+c)*coth(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*arctan((c + I)*coth(b*x + a) + c)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).
time = 2.99, size = 247, normalized size = 2.19 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctan(c+(I+c)*coth(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*dil
og(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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*atan(c+(I+c)*coth(b*x+a)),x)

[Out]

Exception raised: CoercionFailed >> Cannot convert _t0**2*exp(2*a) - 1 of type <class 'sympy.core.add.Add'> to
QQ_I[x,b,_t0,exp(a)]

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctan(c+(I+c)*coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(x*arctan((c + I)*coth(b*x + a) + c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\mathrm {atan}\left (c+\mathrm {coth}\left (a+b\,x\right )\,\left (c+1{}\mathrm {i}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*atan(c + coth(a + b*x)*(c + 1i)),x)

[Out]

int(x*atan(c + coth(a + b*x)*(c + 1i)), x)

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