\(\int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx\) [259]

Optimal result

Integrand size = 16, antiderivative size = 93 \[ \int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {1}{2} i b x^2+x \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{2} x \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {i \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b} \]

[Out]

1/2*I*b*x^2+x*arccoth(1+I*d+d*cot(b*x+a))-1/2*x*ln(1-(1+I*d)*exp(2*I*a+2*I*b*x))+1/4*I*polylog(2,(1+I*d)*exp(2
*I*a+2*I*b*x))/b

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6393, 2215, 2221, 2317, 2438} \[ \int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {i \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{2} x \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+x \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac {1}{2} i b x^2 \]

[In]

Int[ArcCoth[1 + I*d + d*Cot[a + b*x]],x]

[Out]

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

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6393

Int[ArcCoth[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcCoth[c + d*Cot[a + b*x]], x] + Dist
[I*b, Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(709\) vs. \(2(93)=186\).

Time = 2.15 (sec) , antiderivative size = 709, normalized size of antiderivative = 7.62 \[ \int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=x \coth ^{-1}(1+i d+d \cot (a+b x))+\frac {x \csc ^2(a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) (d \cos (a+b x)+(2+i d) \sin (a+b x))}{2 (-i+d)}\right ) \log (1-i \tan (b x))-i \log \left (\frac {1}{2} \sec (b x) (-i \cos (a)+\sin (a)) (d \cos (a+b x)+(2+i d) \sin (a+b x))\right ) \log (1+i \tan (b x))+i \operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))-i \operatorname {PolyLog}\left (2,\frac {1}{2} \sec (b x) ((2+i d) \cos (a)-d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right )+i \operatorname {PolyLog}\left (2,\frac {(\cos (a)-i \sin (a)) ((-2-i d) \cos (a)+d \sin (a)) (i+\tan (b x))}{2 (-i+d)}\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(i+\cot (a+b x)) (2+i d+d \cot (a+b x)) \left (2 i b x+\log \left (1+\frac {1}{2} \sec (b x) ((-2-i d) \cos (a)+d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right )-\log \left (\frac {1}{2} \sec (b x) (-i \cos (a)+\sin (a)) (d \cos (a+b x)+(2+i d) \sin (a+b x))\right )+\frac {(-2 i+d) \cos (a+b x) (\log (1-i \tan (b x))-\log (1+i \tan (b x)))}{d \cos (a+b x)+(2+i d) \sin (a+b x)}+\frac {d (\log (1-i \tan (b x))-\log (1+i \tan (b x))) \sin (a+b x)}{-i d \cos (a+b x)+(-2 i+d) \sin (a+b x)}+2 b x \tan (b x)-i \log \left (1+\frac {1}{2} \sec (b x) ((-2-i d) \cos (a)+d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right ) \tan (b x)+i \log \left (\frac {1}{2} \sec (b x) (-i \cos (a)+\sin (a)) (d \cos (a+b x)+(2+i d) \sin (a+b x))\right ) \tan (b x)-i \log (1-i \tan (b x)) \tan (b x)+i \log (1+i \tan (b x)) \tan (b x)\right )} \]

[In]

Integrate[ArcCoth[1 + I*d + d*Cot[a + b*x]],x]

[Out]

x*ArcCoth[1 + I*d + d*Cot[a + b*x]] + (x*Csc[a + b*x]^2*(2*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log
[(Sec[b*x]*(Cos[a] - I*Sin[a])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/(2*(-I + d))]*Log[1 - I*Tan[b*x]] -
I*Log[(Sec[b*x]*((-I)*Cos[a] + Sin[a])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/2]*Log[1 + I*Tan[b*x]] + I*P
olyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] - I*PolyLog[2, (Sec[b*x]*((2 + I*d)*Cos[a] - d*Sin[a])*(Cos[a + b*x] + I
*Sin[a + b*x]))/2] + I*PolyLog[2, ((Cos[a] - I*Sin[a])*((-2 - I*d)*Cos[a] + d*Sin[a])*(I + Tan[b*x]))/(2*(-I +
d))])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(2 + I*d + d*Cot[a + b*x])*((2*I)*
b*x + Log[1 + (Sec[b*x]*((-2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] - Log[(Sec[b*x]*((-
I)*Cos[a] + Sin[a])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/2] + ((-2*I + d)*Cos[a + b*x]*(Log[1 - I*Tan[b*
x]] - Log[1 + I*Tan[b*x]]))/(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]) + (d*(Log[1 - I*Tan[b*x]] - Log[1 + I*Ta
n[b*x]])*Sin[a + b*x])/((-I)*d*Cos[a + b*x] + (-2*I + d)*Sin[a + b*x]) + 2*b*x*Tan[b*x] - I*Log[1 + (Sec[b*x]*
((-2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] + I*Log[(Sec[b*x]*((-I)*Cos[a] + S
in[a])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/2]*Tan[b*x] - I*Log[1 - I*Tan[b*x]]*Tan[b*x] + I*Log[1 + I*T
an[b*x]]*Tan[b*x]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (76 ) = 152\).

Time = 2.55 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.30

method	result	size
derivativedivides	\(\frac {-\frac {i \operatorname {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d +d \cot \left (b x +a \right )\right )}{2}+\frac {i \operatorname {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (-i d +d \cot \left (b x +a \right )\right )}{2}+\frac {d^{2} \left (-\frac {i \left (-\frac {\operatorname {dilog}\left (1+\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2}-\frac {\ln \left (i d +d \cot \left (b x +a \right )\right ) \ln \left (1+\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2}+\frac {\ln \left (i d +d \cot \left (b x +a \right )\right )^{2}}{4}\right )}{d}+\frac {i \left (\frac {\operatorname {dilog}\left (-\frac {i \left (i d +d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}+\frac {\ln \left (-i d +d \cot \left (b x +a \right )\right ) \ln \left (-\frac {i \left (i d +d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i \left (-i d +d \cot \left (b x +a \right )-i \left (2 i-2 d \right )\right )}{2 i-2 d}\right )}{2}-\frac {\ln \left (-i d +d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (-i d +d \cot \left (b x +a \right )-i \left (2 i-2 d \right )\right )}{2 i-2 d}\right )}{2}\right )}{d}\right )}{2}}{b d}\)	\(307\)
default	\(\frac {-\frac {i \operatorname {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d +d \cot \left (b x +a \right )\right )}{2}+\frac {i \operatorname {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (-i d +d \cot \left (b x +a \right )\right )}{2}+\frac {d^{2} \left (-\frac {i \left (-\frac {\operatorname {dilog}\left (1+\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2}-\frac {\ln \left (i d +d \cot \left (b x +a \right )\right ) \ln \left (1+\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2}+\frac {\ln \left (i d +d \cot \left (b x +a \right )\right )^{2}}{4}\right )}{d}+\frac {i \left (\frac {\operatorname {dilog}\left (-\frac {i \left (i d +d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}+\frac {\ln \left (-i d +d \cot \left (b x +a \right )\right ) \ln \left (-\frac {i \left (i d +d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i \left (-i d +d \cot \left (b x +a \right )-i \left (2 i-2 d \right )\right )}{2 i-2 d}\right )}{2}-\frac {\ln \left (-i d +d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (-i d +d \cot \left (b x +a \right )-i \left (2 i-2 d \right )\right )}{2 i-2 d}\right )}{2}\right )}{d}\right )}{2}}{b d}\)	\(307\)
risch	\(\text {Expression too large to display}\)	\(1650\)

[In]

int(arccoth(1+I*d+d*cot(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/d*(-1/2*I*arccoth(1+I*d+d*cot(b*x+a))*d*ln(I*d+d*cot(b*x+a))+1/2*I*arccoth(1+I*d+d*cot(b*x+a))*d*ln(-I*d+d
*cot(b*x+a))+1/2*d^2*(-I/d*(-1/2*dilog(1+1/2*I*d+1/2*d*cot(b*x+a))-1/2*ln(I*d+d*cot(b*x+a))*ln(1+1/2*I*d+1/2*d
*cot(b*x+a))+1/4*ln(I*d+d*cot(b*x+a))^2)+I/d*(1/2*dilog(-1/2*I*(I*d+d*cot(b*x+a))/d)+1/2*ln(-I*d+d*cot(b*x+a))
*ln(-1/2*I*(I*d+d*cot(b*x+a))/d)-1/2*dilog(I*(-I*d+d*cot(b*x+a)-I*(2*I-2*d))/(2*I-2*d))-1/2*ln(-I*d+d*cot(b*x+
a))*ln(I*(-I*d+d*cot(b*x+a)-I*(2*I-2*d))/(2*I-2*d)))))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30 \[ \int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {2 i \, b^{2} x^{2} + 2 \, b x \log \left (\frac {{\left ({\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{d}\right ) - 2 i \, a^{2} - 2 \, {\left (b x + a\right )} \log \left ({\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 2 \, a \log \left (\frac {{\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i}{d - i}\right ) + i \, {\rm Li}_2\left (-{\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \]

[In]

integrate(arccoth(1+I*d+d*cot(b*x+a)),x, algorithm="fricas")

[Out]

1/4*(2*I*b^2*x^2 + 2*b*x*log(((d - I)*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/d) - 2*I*a^2 - 2*(b*x + a)
*log((-I*d - 1)*e^(2*I*b*x + 2*I*a) + 1) + 2*a*log(((d - I)*e^(2*I*b*x + 2*I*a) + I)/(d - I)) + I*dilog(-(-I*d
- 1)*e^(2*I*b*x + 2*I*a)))/b

Sympy [F]

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

[In]

integrate(acoth(1+I*d+d*cot(b*x+a)),x)

[Out]

Integral(acoth(d*cot(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. 286 vs. \(2 (66) = 132\).

Time = 0.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.08 \[ \int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=-\frac {4 \, {\left (b x + a\right )} d {\left (\frac {\log \left ({\left (i \, d + 2\right )} \tan \left (b x + a\right ) + d\right )}{d} - \frac {\log \left (i \, \tan \left (b x + a\right ) + 1\right )}{d}\right )} + d {\left (-\frac {2 i \, {\left (\log \left ({\left (i \, d + 2\right )} \tan \left (b x + a\right ) + d\right ) \log \left (\frac {{\left (d - 2 i\right )} \tan \left (b x + a\right ) - i \, d}{2 i \, d + 2} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (d - 2 i\right )} \tan \left (b x + a\right ) - i \, d}{2 i \, d + 2}\right )\right )}}{d} - \frac {2 i \, {\left (\log \left (\frac {1}{2} \, {\left (d - 2 i\right )} \tan \left (b x + a\right ) - \frac {1}{2} i \, d\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (-\frac {1}{2} \, {\left (d - 2 i\right )} \tan \left (b x + a\right ) + \frac {1}{2} i \, d + 1\right )\right )}}{d} + \frac {2 i \, \log \left ({\left (i \, d + 2\right )} \tan \left (b x + a\right ) + d\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) - i \, \log \left (i \, \tan \left (b x + a\right ) + 1\right )^{2}}{d} + \frac {2 i \, {\left (\log \left (i \, \tan \left (b x + a\right ) + 1\right ) \log \left (-\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right )\right )}}{d}\right )} - 8 \, {\left (b x + a\right )} \operatorname {arcoth}\left (i \, d + \frac {d}{\tan \left (b x + a\right )} + 1\right )}{8 \, b} \]

[In]

integrate(arccoth(1+I*d+d*cot(b*x+a)),x, algorithm="maxima")

[Out]

-1/8*(4*(b*x + a)*d*(log((I*d + 2)*tan(b*x + a) + d)/d - log(I*tan(b*x + a) + 1)/d) + d*(-2*I*(log((I*d + 2)*t
an(b*x + a) + d)*log(((d - 2*I)*tan(b*x + a) - I*d)/(2*I*d + 2) + 1) + dilog(-((d - 2*I)*tan(b*x + a) - I*d)/(
2*I*d + 2)))/d - 2*I*(log(1/2*(d - 2*I)*tan(b*x + a) - 1/2*I*d)*log(I*tan(b*x + a) + 1) + dilog(-1/2*(d - 2*I)
*tan(b*x + a) + 1/2*I*d + 1))/d + (2*I*log((I*d + 2)*tan(b*x + a) + d)*log(I*tan(b*x + a) + 1) - I*log(I*tan(b
*x + a) + 1)^2)/d + 2*I*(log(I*tan(b*x + a) + 1)*log(-1/2*I*tan(b*x + a) + 1/2) + dilog(1/2*I*tan(b*x + a) + 1
/2))/d) - 8*(b*x + a)*arccoth(I*d + d/tan(b*x + a) + 1))/b

Giac [F]

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

[In]

integrate(arccoth(1+I*d+d*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(arccoth(d*cot(b*x + a) + I*d + 1), x)

Mupad [F(-1)]

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

[In]

int(acoth(d*1i + d*cot(a + b*x) + 1),x)

[Out]

int(acoth(d*1i + d*cot(a + b*x) + 1), x)