3.4.36 \(\int \tanh ^{-1}(c+d \cot (a+b x)) \, dx\) [336]

Optimal. Leaf size=194 \[ x \tanh ^{-1}(c+d \cot (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )-\frac {i \text {PolyLog}\left (2,\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )}{4 b}+\frac {i \text {PolyLog}\left (2,\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )}{4 b} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6396, 2221, 2317, 2438} \begin {gather*} -\frac {i \text {Li}_2\left (\frac {(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right )}{4 b}+\frac {i \text {Li}_2\left (\frac {(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right )}{4 b}+\frac {1}{2} x \log \left (1-\frac {(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right )-\frac {1}{2} x \log \left (1-\frac {(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right )+x \tanh ^{-1}(d \cot (a+b x)+c) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[c + d*Cot[a + b*x]],x]

[Out]

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

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 6396

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

Rubi steps

\begin {align*} \int \tanh ^{-1}(c+d \cot (a+b x)) \, dx &=x \tanh ^{-1}(c+d \cot (a+b x))+(b (i (1+c)-d)) \int \frac {e^{2 i a+2 i b x} x}{1+c-i d+(-1-c-i d) e^{2 i a+2 i b x}} \, dx-(b (i-i c+d)) \int \frac {e^{2 i a+2 i b x} x}{1-c+i d+(-1+c+i d) e^{2 i a+2 i b x}} \, dx\\ &=x \tanh ^{-1}(c+d \cot (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )+\frac {1}{2} \int \log \left (1+\frac {(-1-c-i d) e^{2 i a+2 i b x}}{1+c-i d}\right ) \, dx-\frac {1}{2} \int \log \left (1+\frac {(-1+c+i d) e^{2 i a+2 i b x}}{1-c+i d}\right ) \, dx\\ &=x \tanh ^{-1}(c+d \cot (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {(-1-c-i d) x}{1+c-i d}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {(-1+c+i d) x}{1-c+i d}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b}\\ &=x \tanh ^{-1}(c+d \cot (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )-\frac {i \text {Li}_2\left (\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )}{4 b}+\frac {i \text {Li}_2\left (\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )}{4 b}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4463\) vs. \(2(194)=388\).
time = 29.49, size = 4463, normalized size = 23.01 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[c + d*Cot[a + b*x]],x]

[Out]

x*ArcTanh[c + d*Cot[a + b*x]] - (d*(a*Log[-(Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] + (-1 + c)*Sin[a + b*x]))] - a*
Log[-(Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] + Sin[a + b*x] + c*Sin[a + b*x]))] - (a + b*x)*Log[-((-1 + c + Sqrt[1
- 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - I*Log[(d*(-I + Tan[(a + b*x)/2]))/(-1 + c - I*d + Sqrt[1 - 2*c +
c^2 + d^2])]*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] + I*Log[(d*(I + Tan[(a + b*x)/
2]))/(-1 + c + I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)
/2]] + (a + b*x)*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] + I*Log[(d*(-I + Tan[(a + b*
x)/2]))/(1 + c - I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x
)/2]] - I*Log[(d*(I + Tan[(a + b*x)/2]))/(1 + c + I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((1 + c + Sqrt[1 + 2*
c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - (a + b*x)*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]
)/d] - I*Log[-((d*(-I + Tan[(a + b*x)/2]))/(1 - c + I*d + Sqrt[1 - 2*c + c^2 + d^2]))]*Log[(1 - c + Sqrt[1 - 2
*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + I*Log[-((d*(I + Tan[(a + b*x)/2]))/(1 - c - I*d + Sqrt[1 - 2*c + c^
2 + d^2]))]*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + (a + b*x)*Log[(-1 - c + Sqrt[1 +
2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + I*Log[-((d*(-I + Tan[(a + b*x)/2]))/(-1 - c + I*d + Sqrt[1 + 2*c
+ c^2 + d^2]))]*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] - I*Log[-((d*(I + Tan[(a + b*
x)/2]))/(-1 - c - I*d + Sqrt[1 + 2*c + c^2 + d^2]))]*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)
/2])/d] - I*PolyLog[2, (-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(-1 + c - I*d + Sqrt[1 - 2*c
+ c^2 + d^2])] + I*PolyLog[2, (-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(-1 + c + I*d + Sqrt[1
- 2*c + c^2 + d^2])] - I*PolyLog[2, (1 + c - Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c + I*d - S
qrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c - I*
d + Sqrt[1 + 2*c + c^2 + d^2])] - I*PolyLog[2, (1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c
+ I*d + Sqrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(
1 - c - I*d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[2, (1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/
2])/(1 - c + I*d + Sqrt[1 - 2*c + c^2 + d^2])] + I*PolyLog[2, (-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a +
b*x)/2])/(-1 - c + I*d + Sqrt[1 + 2*c + c^2 + d^2])])*((2*a)/(b*(1 - c^2 - d^2 - Cos[2*(a + b*x)] + c^2*Cos[2
*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)])) - (2*(a + b*x))/(b*(1 - c^2 - d^2 - Cos[2*(a + b
*x)] + c^2*Cos[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)]))))/(-Log[-((-1 + c + Sqrt[1 - 2*c
+ c^2 + d^2])/d) + Tan[(a + b*x)/2]] + Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - Log
[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan
[(a + b*x)/2])/d] - ((I/2)*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^
2)/(-I + Tan[(a + b*x)/2]) + ((I/2)*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]]*Sec[(a +
b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) - ((I/2)*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d]*Se
c[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) + ((I/2)*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2
])/d]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) + ((I/2)*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan
[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((I/2)*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d
) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) + ((I/2)*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^
2] + d*Tan[(a + b*x)/2])/d]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((I/2)*Log[(-1 - c + Sqrt[1 + 2*c + c
^2 + d^2] + d*Tan[(a + b*x)/2])/d]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((a + b*x)*Sec[(a + b*x)/2]^2)
/(2*(-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2])) - ((I/2)*Log[(d*(-I + Tan[(a + b*x)/2]))/(
-1 + c - I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Ta
n[(a + b*x)/2]) + ((I/2)*Log[(d*(I + Tan[(a + b*x)/2]))/(-1 + c + I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b
*x)/2]^2)/(-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]) + ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*(
-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2])) + ((I/2)*Log[(d*(-I + Tan[(a + b*x)/2]))/(1 + c
- I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b
*x)/2]) - ((I/2)*Log[(d*(I + Tan[(a + b*x)/2]))/(1 + c + I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)
/(-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]) - ((I/2)*d*Log[1 - (-1 + c + Sqrt[1 - 2*c + c^2
+ d^2] - d*Tan[(a + b*x)/2])/(-1 + c - I*d + S...

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (164 ) = 328\).
time = 1.36, size = 569, normalized size = 2.93

method	result	size
derivativedivides	\(\frac {-d \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )\right ) \arctanh \left (c +d \cot \left (b x +a \right )\right )+d^{2} \left (-\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right )}{2 d}+\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right )}{2 d}+\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}+\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )}{4 d}-\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )}{4 d}-\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}-\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )}{4 d}+\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )}{4 d}\right )}{b d}\)	\(569\)
default	\(\frac {-d \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )\right ) \arctanh \left (c +d \cot \left (b x +a \right )\right )+d^{2} \left (-\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right )}{2 d}+\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right )}{2 d}+\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}+\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )}{4 d}-\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )}{4 d}-\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}-\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )}{4 d}+\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )}{4 d}\right )}{b d}\)	\(569\)
risch	\(\text {Expression too large to display}\)	\(3982\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(c+d*cot(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/d*(-d*(1/2*Pi-arccot(cot(b*x+a)))*arctanh(c+d*cot(b*x+a))+d^2*(-1/2*arctan(-(c+d*cot(b*x+a))/d+c/d)/d*ln(d
*((c+d*cot(b*x+a))/d-c/d)+c+1)+1/2*arctan(-(c+d*cot(b*x+a))/d+c/d)/d*ln(d*((c+d*cot(b*x+a))/d-c/d)+c-1)+1/4*I*
ln(d*((c+d*cot(b*x+a))/d-c/d)+c+1)*(ln((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(1+c+I*d))-ln((I*d+d*((c+d*cot(b*x+a))
/d-c/d))/(I*d-c-1)))/d+1/4*I/d*dilog((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(1+c+I*d))-1/4*I/d*dilog((I*d+d*((c+d*co
t(b*x+a))/d-c/d))/(I*d-c-1))-1/4*I*ln(d*((c+d*cot(b*x+a))/d-c/d)+c-1)*(ln((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(I*
d+c-1))-ln((I*d+d*((c+d*cot(b*x+a))/d-c/d))/(1-c+I*d)))/d-1/4*I/d*dilog((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(I*d+
c-1))+1/4*I/d*dilog((I*d+d*((c+d*cot(b*x+a))/d-c/d))/(1-c+I*d))))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (136) = 272\).
time = 0.53, size = 392, normalized size = 2.02 \begin {gather*} \frac {4 \, {\left (b x + a\right )} \operatorname {artanh}\left (c + \frac {d}{\tan \left (b x + a\right )}\right ) + {\left (\arctan \left (\frac {{\left (c + 1\right )} d + {\left (c^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} + 2 \, c + 1}, \frac {{\left (c + 1\right )} d \tan \left (b x + a\right ) + d^{2}}{c^{2} + d^{2} + 2 \, c + 1}\right ) - \arctan \left (\frac {{\left (c - 1\right )} d + {\left (c^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} - 2 \, c + 1}, \frac {{\left (c - 1\right )} d \tan \left (b x + a\right ) + d^{2}}{c^{2} + d^{2} - 2 \, c + 1}\right )\right )} \log \left (\tan \left (b x + a\right )^{2} + 1\right ) - {\left (b x + a\right )} \log \left (\frac {2 \, {\left (c + 1\right )} d \tan \left (b x + a\right ) + {\left (c^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + d^{2}}{c^{2} + d^{2} + 2 \, c + 1}\right ) + {\left (b x + a\right )} \log \left (\frac {2 \, {\left (c - 1\right )} d \tan \left (b x + a\right ) + {\left (c^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + d^{2}}{c^{2} + d^{2} - 2 \, c + 1}\right ) + i \, {\rm Li}_2\left (-\frac {{\left (c + 1\right )} \tan \left (b x + a\right ) - i \, c - i}{i \, c + d + i}\right ) - i \, {\rm Li}_2\left (-\frac {{\left (c - 1\right )} \tan \left (b x + a\right ) - i \, c + i}{i \, c + d - i}\right ) + i \, {\rm Li}_2\left (-\frac {{\left (c - 1\right )} \tan \left (b x + a\right ) + i \, c - i}{-i \, c + d + i}\right ) - i \, {\rm Li}_2\left (-\frac {{\left (c + 1\right )} \tan \left (b x + a\right ) + i \, c + i}{-i \, c + d - i}\right )}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(c+d*cot(b*x+a)),x, algorithm="maxima")

[Out]

1/4*(4*(b*x + a)*arctanh(c + d/tan(b*x + a)) + (arctan2(((c + 1)*d + (c^2 + 2*c + 1)*tan(b*x + a))/(c^2 + d^2
+ 2*c + 1), ((c + 1)*d*tan(b*x + a) + d^2)/(c^2 + d^2 + 2*c + 1)) - arctan2(((c - 1)*d + (c^2 - 2*c + 1)*tan(b
*x + a))/(c^2 + d^2 - 2*c + 1), ((c - 1)*d*tan(b*x + a) + d^2)/(c^2 + d^2 - 2*c + 1)))*log(tan(b*x + a)^2 + 1)
- (b*x + a)*log((2*(c + 1)*d*tan(b*x + a) + (c^2 + 2*c + 1)*tan(b*x + a)^2 + d^2)/(c^2 + d^2 + 2*c + 1)) + (b
*x + a)*log((2*(c - 1)*d*tan(b*x + a) + (c^2 - 2*c + 1)*tan(b*x + a)^2 + d^2)/(c^2 + d^2 - 2*c + 1)) + I*dilog
(-((c + 1)*tan(b*x + a) - I*c - I)/(I*c + d + I)) - I*dilog(-((c - 1)*tan(b*x + a) - I*c + I)/(I*c + d - I)) +
I*dilog(-((c - 1)*tan(b*x + a) + I*c - I)/(-I*c + d + I)) - I*dilog(-((c + 1)*tan(b*x + a) + I*c + I)/(-I*c +
d - I)))/b

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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. 1099 vs. \(2 (136) = 272\).
time = 0.59, size = 1099, normalized size = 5.66 \begin {gather*} \frac {4 \, b x \log \left (-\frac {d \cos \left (2 \, b x + 2 \, a\right ) + {\left (c + 1\right )} \sin \left (2 \, b x + 2 \, a\right ) + d}{d \cos \left (2 \, b x + 2 \, a\right ) + {\left (c - 1\right )} \sin \left (2 \, b x + 2 \, a\right ) + d}\right ) + 2 \, a \log \left (\frac {1}{2} \, c^{2} + i \, {\left (c + 1\right )} d - \frac {1}{2} \, d^{2} - \frac {1}{2} \, {\left (c^{2} + d^{2} + 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} + 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + c + \frac {1}{2}\right ) - 2 \, a \log \left (\frac {1}{2} \, c^{2} + i \, {\left (c - 1\right )} d - \frac {1}{2} \, d^{2} - \frac {1}{2} \, {\left (c^{2} + d^{2} - 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} - 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - c + \frac {1}{2}\right ) + 2 \, a \log \left (-\frac {1}{2} \, c^{2} + i \, {\left (c + 1\right )} d + \frac {1}{2} \, d^{2} + \frac {1}{2} \, {\left (c^{2} + d^{2} + 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} + 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - c - \frac {1}{2}\right ) - 2 \, a \log \left (-\frac {1}{2} \, c^{2} + i \, {\left (c - 1\right )} d + \frac {1}{2} \, d^{2} + \frac {1}{2} \, {\left (c^{2} + d^{2} - 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, {\left (i \, c^{2} + i \, d^{2} - 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + c - \frac {1}{2}\right ) - 2 \, {\left (b x + a\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, {\left (c + 1\right )} d - d^{2} + 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, {\left (c + 1\right )} d + i \, d^{2} - 2 i \, c - i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, c + 1}{c^{2} + d^{2} + 2 \, c + 1}\right ) - 2 \, {\left (b x + a\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, {\left (c + 1\right )} d - d^{2} + 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, {\left (c + 1\right )} d - i \, d^{2} + 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, c + 1}{c^{2} + d^{2} + 2 \, c + 1}\right ) + 2 \, {\left (b x + a\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, {\left (c - 1\right )} d - d^{2} - 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, {\left (c - 1\right )} d + i \, d^{2} + 2 i \, c - i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, c + 1}{c^{2} + d^{2} - 2 \, c + 1}\right ) + 2 \, {\left (b x + a\right )} \log \left (\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, {\left (c - 1\right )} d - d^{2} - 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, {\left (c - 1\right )} d - i \, d^{2} - 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, c + 1}{c^{2} + d^{2} - 2 \, c + 1}\right ) + i \, {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, {\left (c + 1\right )} d - d^{2} + 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, {\left (c + 1\right )} d + i \, d^{2} - 2 i \, c - i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, c + 1}{c^{2} + d^{2} + 2 \, c + 1} + 1\right ) - i \, {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, {\left (c + 1\right )} d - d^{2} + 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, {\left (c + 1\right )} d - i \, d^{2} + 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, c + 1}{c^{2} + d^{2} + 2 \, c + 1} + 1\right ) - i \, {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} + 2 i \, {\left (c - 1\right )} d - d^{2} - 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, c^{2} + 2 \, {\left (c - 1\right )} d + i \, d^{2} + 2 i \, c - i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, c + 1}{c^{2} + d^{2} - 2 \, c + 1} + 1\right ) + i \, {\rm Li}_2\left (-\frac {c^{2} + d^{2} - {\left (c^{2} - 2 i \, {\left (c - 1\right )} d - d^{2} - 2 \, c + 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (i \, c^{2} + 2 \, {\left (c - 1\right )} d - i \, d^{2} - 2 i \, c + i\right )} \sin \left (2 \, b x + 2 \, a\right ) - 2 \, c + 1}{c^{2} + d^{2} - 2 \, c + 1} + 1\right )}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(c+d*cot(b*x+a)),x, algorithm="fricas")

[Out]

1/8*(4*b*x*log(-(d*cos(2*b*x + 2*a) + (c + 1)*sin(2*b*x + 2*a) + d)/(d*cos(2*b*x + 2*a) + (c - 1)*sin(2*b*x +
2*a) + d)) + 2*a*log(1/2*c^2 + I*(c + 1)*d - 1/2*d^2 - 1/2*(c^2 + d^2 + 2*c + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2
+ I*d^2 + 2*I*c + I)*sin(2*b*x + 2*a) + c + 1/2) - 2*a*log(1/2*c^2 + I*(c - 1)*d - 1/2*d^2 - 1/2*(c^2 + d^2 -
2*c + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 - 2*I*c + I)*sin(2*b*x + 2*a) - c + 1/2) + 2*a*log(-1/2*c^2 +
I*(c + 1)*d + 1/2*d^2 + 1/2*(c^2 + d^2 + 2*c + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 + 2*I*c + I)*sin(2*b*x
+ 2*a) - c - 1/2) - 2*a*log(-1/2*c^2 + I*(c - 1)*d + 1/2*d^2 + 1/2*(c^2 + d^2 - 2*c + 1)*cos(2*b*x + 2*a) + 1
/2*(I*c^2 + I*d^2 - 2*I*c + I)*sin(2*b*x + 2*a) + c - 1/2) - 2*(b*x + a)*log((c^2 + d^2 - (c^2 + 2*I*(c + 1)*d
- d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*(c + 1)*d + I*d^2 - 2*I*c - I)*sin(2*b*x + 2*a) + 2*c + 1)/(c
^2 + d^2 + 2*c + 1)) - 2*(b*x + a)*log((c^2 + d^2 - (c^2 - 2*I*(c + 1)*d - d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (
I*c^2 + 2*(c + 1)*d - I*d^2 + 2*I*c + I)*sin(2*b*x + 2*a) + 2*c + 1)/(c^2 + d^2 + 2*c + 1)) + 2*(b*x + a)*log(
(c^2 + d^2 - (c^2 + 2*I*(c - 1)*d - d^2 - 2*c + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*(c - 1)*d + I*d^2 + 2*I*c -
I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^2 - 2*c + 1)) + 2*(b*x + a)*log((c^2 + d^2 - (c^2 - 2*I*(c - 1)*d - d^
2 - 2*c + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*(c - 1)*d - I*d^2 - 2*I*c + I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d
^2 - 2*c + 1)) + I*dilog(-(c^2 + d^2 - (c^2 + 2*I*(c + 1)*d - d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*(c
+ 1)*d + I*d^2 - 2*I*c - I)*sin(2*b*x + 2*a) + 2*c + 1)/(c^2 + d^2 + 2*c + 1) + 1) - I*dilog(-(c^2 + d^2 - (c
^2 - 2*I*(c + 1)*d - d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*(c + 1)*d - I*d^2 + 2*I*c + I)*sin(2*b*x + 2
*a) + 2*c + 1)/(c^2 + d^2 + 2*c + 1) + 1) - I*dilog(-(c^2 + d^2 - (c^2 + 2*I*(c - 1)*d - d^2 - 2*c + 1)*cos(2*
b*x + 2*a) + (-I*c^2 + 2*(c - 1)*d + I*d^2 + 2*I*c - I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^2 - 2*c + 1) + 1)
+ I*dilog(-(c^2 + d^2 - (c^2 - 2*I*(c - 1)*d - d^2 - 2*c + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*(c - 1)*d - I*d^2
- 2*I*c + I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^2 - 2*c + 1) + 1))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {atanh}{\left (c + d \cot {\left (a + b x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(c+d*cot(b*x+a)),x)

[Out]

Integral(atanh(c + d*cot(a + b*x)), x)

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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(arctanh(c+d*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(arctanh(d*cot(b*x + a) + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(c + d*cot(a + b*x)),x)

[Out]

int(atanh(c + d*cot(a + b*x)), x)

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