3.3.100 \(\int \tanh ^{-1}(c+d \coth (a+b x)) \, dx\) [300]

Optimal. Leaf size=150 \[ x \tanh ^{-1}(c+d \coth (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac {\text {PolyLog}\left (2,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac {\text {PolyLog}\left (2,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 150, 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 = {6372, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac {\text {Li}_2\left (\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac {1}{2} x \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )-\frac {1}{2} x \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )+x \tanh ^{-1}(d \coth (a+b x)+c) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[c + d*Coth[a + b*x]],x]

[Out]

x*ArcTanh[c + d*Coth[a + b*x]] + (x*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/2 - (x*Log[1 - ((1 + c
+ d)*E^(2*a + 2*b*x))/(1 + c - d)])/2 + PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(4*b) - PolyLog
[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(4*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 6372

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(369\) vs. \(2(150)=300\).
time = 3.33, size = 369, normalized size = 2.46 \begin {gather*} x \tanh ^{-1}(c+d \coth (a+b x))-\frac {-\left ((a+b x) \log \left (1-\frac {\sqrt {-1+c+d} e^{a+b x}}{\sqrt {-1+c-d}}\right )\right )-(a+b x) \log \left (1+\frac {\sqrt {-1+c+d} e^{a+b x}}{\sqrt {-1+c-d}}\right )+(a+b x) \log \left (1-\frac {\sqrt {1+c+d} e^{a+b x}}{\sqrt {1+c-d}}\right )+(a+b x) \log \left (1+\frac {\sqrt {1+c+d} e^{a+b x}}{\sqrt {1+c-d}}\right )+a \log \left (1+d-e^{2 (a+b x)}+d e^{2 (a+b x)}+c \left (-1+e^{2 (a+b x)}\right )\right )-a \log \left (1+c-e^{2 (a+b x)}-c e^{2 (a+b x)}-d \left (1+e^{2 (a+b x)}\right )\right )-\text {PolyLog}\left (2,-\frac {\sqrt {-1+c+d} e^{a+b x}}{\sqrt {-1+c-d}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {-1+c+d} e^{a+b x}}{\sqrt {-1+c-d}}\right )+\text {PolyLog}\left (2,-\frac {\sqrt {1+c+d} e^{a+b x}}{\sqrt {1+c-d}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {1+c+d} e^{a+b x}}{\sqrt {1+c-d}}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*ArcTanh[c + d*Coth[a + b*x]] - (-((a + b*x)*Log[1 - (Sqrt[-1 + c + d]*E^(a + b*x))/Sqrt[-1 + c - d]]) - (a +
b*x)*Log[1 + (Sqrt[-1 + c + d]*E^(a + b*x))/Sqrt[-1 + c - d]] + (a + b*x)*Log[1 - (Sqrt[1 + c + d]*E^(a + b*x
))/Sqrt[1 + c - d]] + (a + b*x)*Log[1 + (Sqrt[1 + c + d]*E^(a + b*x))/Sqrt[1 + c - d]] + a*Log[1 + d - E^(2*(a
+ b*x)) + d*E^(2*(a + b*x)) + c*(-1 + E^(2*(a + b*x)))] - a*Log[1 + c - E^(2*(a + b*x)) - c*E^(2*(a + b*x)) -
d*(1 + E^(2*(a + b*x)))] - PolyLog[2, -((Sqrt[-1 + c + d]*E^(a + b*x))/Sqrt[-1 + c - d])] - PolyLog[2, (Sqrt[
-1 + c + d]*E^(a + b*x))/Sqrt[-1 + c - d]] + PolyLog[2, -((Sqrt[1 + c + d]*E^(a + b*x))/Sqrt[1 + c - d])] + Po
lyLog[2, (Sqrt[1 + c + d]*E^(a + b*x))/Sqrt[1 + c - d]])/(2*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(138)=276\).
time = 1.02, size = 361, normalized size = 2.41

method	result	size
derivativedivides	\(\frac {\frac {\arctanh \left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )-d \right )}{2}-\frac {\arctanh \left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )+d \right )}{2}+\frac {d^{2} \left (-\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2 d}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2 d}+\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2 d}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2 d}-\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2 d}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2 d}+\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2 d}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2 d}\right )}{2}}{b d}\)	\(361\)
default	\(\frac {\frac {\arctanh \left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )-d \right )}{2}-\frac {\arctanh \left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )+d \right )}{2}+\frac {d^{2} \left (-\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2 d}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2 d}+\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2 d}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2 d}-\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2 d}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2 d}+\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2 d}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2 d}\right )}{2}}{b d}\)	\(361\)
risch	\(\text {Expression too large to display}\)	\(3019\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/d*(1/2*arctanh(c+d*coth(b*x+a))*d*ln(-d*coth(b*x+a)-d)-1/2*arctanh(c+d*coth(b*x+a))*d*ln(-d*coth(b*x+a)+d)
+1/2*d^2*(-1/2/d*dilog((-d*coth(b*x+a)-c-1)/(-1-c+d))-1/2/d*ln(-d*coth(b*x+a)-d)*ln((-d*coth(b*x+a)-c-1)/(-1-c
+d))+1/2/d*dilog((-d*coth(b*x+a)-c+1)/(1-c+d))+1/2/d*ln(-d*coth(b*x+a)-d)*ln((-d*coth(b*x+a)-c+1)/(1-c+d))-1/2
/d*dilog((-d*coth(b*x+a)-c+1)/(1-c-d))-1/2/d*ln(-d*coth(b*x+a)+d)*ln((-d*coth(b*x+a)-c+1)/(1-c-d))+1/2/d*dilog
((-d*coth(b*x+a)-c-1)/(-1-c-d))+1/2/d*ln(-d*coth(b*x+a)+d)*ln((-d*coth(b*x+a)-c-1)/(-1-c-d))))

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Maxima [A]
time = 0.49, size = 142, normalized size = 0.95 \begin {gather*} -\frac {1}{4} \, b d {\left (\frac {2 \, b x \log \left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + {\rm Li}_2\left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right )}{b^{2} d} - \frac {2 \, b x \log \left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + {\rm Li}_2\left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right )}{b^{2} d}\right )} + x \operatorname {artanh}\left (d \coth \left (b x + a\right ) + c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (128) = 256\).
time = 0.48, size = 540, normalized size = 3.60 \begin {gather*} \frac {b x \log \left (-\frac {d \cosh \left (b x + a\right ) + {\left (c + 1\right )} \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + {\left (c - 1\right )} \sinh \left (b x + a\right )}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d + 1\right )} \sqrt {\frac {c + d + 1}{c - d + 1}}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d + 1\right )} \sqrt {\frac {c + d + 1}{c - d + 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d - 1\right )} \sqrt {\frac {c + d - 1}{c - d - 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d - 1\right )} \sqrt {\frac {c + d - 1}{c - d - 1}}\right ) - {\left (b x + a\right )} \log \left (\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b x + a\right )} \log \left (-\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (-\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\rm Li}_2\left (\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - {\rm Li}_2\left (-\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (-\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x*log(-(d*cosh(b*x + a) + (c + 1)*sinh(b*x + a))/(d*cosh(b*x + a) + (c - 1)*sinh(b*x + a))) + a*log(2*(
c + d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) + 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) + a*log(
2*(c + d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) - 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) - a*l
og(2*(c + d - 1)*cosh(b*x + a) + 2*(c + d - 1)*sinh(b*x + a) + 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1))) -
a*log(2*(c + d - 1)*cosh(b*x + a) + 2*(c + d - 1)*sinh(b*x + a) - 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1)))
- (b*x + a)*log(sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) - (b*x + a)*log(-sqrt((c +
d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (b*x + a)*log(sqrt((c + d - 1)/(c - d - 1))*(cosh(
b*x + a) + sinh(b*x + a)) + 1) + (b*x + a)*log(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))
+ 1) - dilog(sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a))) - dilog(-sqrt((c + d + 1)/(c - d +
1))*(cosh(b*x + a) + sinh(b*x + a))) + dilog(sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))) +
dilog(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atanh(c + d*coth(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*coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(arctanh(d*coth(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 {coth}\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*coth(a + b*x)),x)

[Out]

int(atanh(c + d*coth(a + b*x)), x)

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