3.4.49 \(\int \tanh ^{-1}(e^{a+b x}) \, dx\) [349]

Optimal. Leaf size=35 \[ -\frac {\text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b}+\frac {\text {PolyLog}\left (2,e^{a+b x}\right )}{2 b} \]

[Out]

-1/2*polylog(2,-exp(b*x+a))/b+1/2*polylog(2,exp(b*x+a))/b

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 6031} \begin {gather*} \frac {\text {Li}_2\left (e^{a+b x}\right )}{2 b}-\frac {\text {Li}_2\left (-e^{a+b x}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[E^(a + b*x)],x]

[Out]

-1/2*PolyLog[2, -E^(a + b*x)]/b + PolyLog[2, E^(a + b*x)]/(2*b)

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 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps

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

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Mathematica [A]
time = 0.15, size = 69, normalized size = 1.97 \begin {gather*} x \tanh ^{-1}\left (e^{a+b x}\right )+\frac {b x \left (\log \left (1-e^{a+b x}\right )-\log \left (1+e^{a+b x}\right )\right )-\text {PolyLog}\left (2,-e^{a+b x}\right )+\text {PolyLog}\left (2,e^{a+b x}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[E^(a + b*x)],x]

[Out]

x*ArcTanh[E^(a + b*x)] + (b*x*(Log[1 - E^(a + b*x)] - Log[1 + E^(a + b*x)]) - PolyLog[2, -E^(a + b*x)] + PolyL
og[2, E^(a + b*x)])/(2*b)

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Maple [A]
time = 0.06, size = 59, normalized size = 1.69

method	result	size
risch	\(-\frac {\dilog \left ({\mathrm e}^{b x +a}+1\right )}{2 b}+\frac {\dilog \left (1-{\mathrm e}^{b x +a}\right )}{2 b}\)	\(32\)
derivativedivides	\(\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {\dilog \left ({\mathrm e}^{b x +a}\right )}{2}-\frac {\dilog \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}}{b}\)	\(59\)
default	\(\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {\dilog \left ({\mathrm e}^{b x +a}\right )}{2}-\frac {\dilog \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}}{b}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(exp(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b*(ln(exp(b*x+a))*arctanh(exp(b*x+a))-1/2*dilog(exp(b*x+a))-1/2*dilog(exp(b*x+a)+1)-1/2*ln(exp(b*x+a))*ln(ex
p(b*x+a)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (27) = 54\).
time = 0.26, size = 107, normalized size = 3.06 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {artanh}\left (e^{\left (b x + a\right )}\right )}{b} - \frac {{\left (b x + a\right )} {\left (\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} - 1\right )\right )} - \log \left (-e^{\left (b x + a\right )}\right ) \log \left (e^{\left (b x + a\right )} + 1\right ) + {\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - {\rm Li}_2\left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )} + 1\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(exp(b*x+a)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (27) = 54\).
time = 0.37, size = 138, normalized size = 3.94 \begin {gather*} \frac {b x \log \left (-\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(exp(b*x+a)),x, algorithm="fricas")

[Out]

1/2*(b*x*log(-(cosh(b*x + a) + sinh(b*x + a) + 1)/(cosh(b*x + a) + sinh(b*x + a) - 1)) - b*x*log(cosh(b*x + a)
+ sinh(b*x + a) + 1) - a*log(cosh(b*x + a) + sinh(b*x + a) - 1) + (b*x + a)*log(-cosh(b*x + a) - sinh(b*x + a
) + 1) + dilog(cosh(b*x + a) + sinh(b*x + a)) - dilog(-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 (e^{a + b x} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(exp(b*x+a)),x)

[Out]

Integral(atanh(exp(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(exp(b*x+a)),x, algorithm="giac")

[Out]

integrate(arctanh(e^(b*x + a)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {atanh}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(exp(a + b*x)),x)

[Out]

int(atanh(exp(a + b*x)), x)

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