3.8.5 \(\int e^x \sin ^{-1}(\tanh (x)) \, dx\) [705]

Optimal. Leaf size=28 \[ e^x \sin ^{-1}(\tanh (x))-\cosh (x) \log \left (1+e^{2 x}\right ) \sqrt {\text {sech}^2(x)} \]

[Out]

exp(x)*arcsin(tanh(x))-cosh(x)*ln(1+exp(2*x))*(sech(x)^2)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2225, 4928, 6852, 2320, 12, 266} \begin {gather*} e^x \text {ArcSin}(\tanh (x))-\log \left (e^{2 x}+1\right ) \cosh (x) \sqrt {\text {sech}^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*ArcSin[Tanh[x]],x]

[Out]

E^x*ArcSin[Tanh[x]] - Cosh[x]*Log[1 + E^(2*x)]*Sqrt[Sech[x]^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 4928

Int[((a_.) + ArcSin[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcSin[u], w, x] - Dist
[b, Int[SimplifyIntegrand[w*(D[u, x]/Sqrt[1 - u^2]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}
, x] && InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]

Rule 6852

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m)^FracPart[p]/v^(m*FracPart[p])), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

\begin {align*} \int e^x \sin ^{-1}(\tanh (x)) \, dx &=e^x \sin ^{-1}(\tanh (x))-\int e^x \sqrt {\text {sech}^2(x)} \, dx\\ &=e^x \sin ^{-1}(\tanh (x))-\left (\cosh (x) \sqrt {\text {sech}^2(x)}\right ) \int e^x \text {sech}(x) \, dx\\ &=e^x \sin ^{-1}(\tanh (x))-\left (\cosh (x) \sqrt {\text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {2 x}{1+x^2} \, dx,x,e^x\right )\\ &=e^x \sin ^{-1}(\tanh (x))-\left (2 \cosh (x) \sqrt {\text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,e^x\right )\\ &=e^x \sin ^{-1}(\tanh (x))-\cosh (x) \log \left (1+e^{2 x}\right ) \sqrt {\text {sech}^2(x)}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(28)=56\).
time = 0.56, size = 64, normalized size = 2.29 \begin {gather*} e^x \sin ^{-1}\left (\frac {-1+e^{2 x}}{1+e^{2 x}}\right )-e^{-x} \sqrt {\frac {e^{2 x}}{\left (1+e^{2 x}\right )^2}} \left (1+e^{2 x}\right ) \log \left (1+e^{2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*ArcSin[Tanh[x]],x]

[Out]

E^x*ArcSin[(-1 + E^(2*x))/(1 + E^(2*x))] - (Sqrt[E^(2*x)/(1 + E^(2*x))^2]*(1 + E^(2*x))*Log[1 + E^(2*x)])/E^x

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{x} \arcsin \left (\tanh \left (x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*arcsin(tanh(x)),x)

[Out]

int(exp(x)*arcsin(tanh(x)),x)

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Maxima [A]
time = 1.69, size = 16, normalized size = 0.57 \begin {gather*} \arcsin \left (\tanh \left (x\right )\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*arcsin(tanh(x)),x, algorithm="maxima")

[Out]

arcsin(tanh(x))*e^x - log(e^(2*x) + 1)

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Fricas [A]
time = 0.45, size = 26, normalized size = 0.93 \begin {gather*} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\sinh \left (x\right )\right ) - \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*arcsin(tanh(x)),x, algorithm="fricas")

[Out]

(cosh(x) + sinh(x))*arctan(sinh(x)) - log(2*cosh(x)/(cosh(x) - sinh(x)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \operatorname {asin}{\left (\tanh {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*asin(tanh(x)),x)

[Out]

Integral(exp(x)*asin(tanh(x)), x)

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Giac [A]
time = 1.05, size = 29, normalized size = 1.04 \begin {gather*} \arcsin \left (\frac {e^{\left (2 \, x\right )} - 1}{e^{\left (2 \, x\right )} + 1}\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*arcsin(tanh(x)),x, algorithm="giac")

[Out]

arcsin((e^(2*x) - 1)/(e^(2*x) + 1))*e^x - log(e^(2*x) + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \mathrm {asin}\left (\mathrm {tanh}\left (x\right )\right )\,{\mathrm {e}}^x \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(tanh(x))*exp(x),x)

[Out]

int(asin(tanh(x))*exp(x), x)

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