3.5.37 \(\int e^x \text {ArcSin}(e^x) \, dx\) [437]

Optimal. Leaf size=22 \[ \sqrt {1-e^{2 x}}+e^x \text {ArcSin}\left (e^x\right ) \]

[Out]

exp(x)*arcsin(exp(x))+(1-exp(2*x))^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2225, 4928, 2278, 32} \begin {gather*} e^x \text {ArcSin}\left (e^x\right )+\sqrt {1-e^{2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -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 2278

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \sqrt {1-e^{2 x}}+e^x \text {ArcSin}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]

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Maple [A]
time = 0.02, size = 18, normalized size = 0.82

method	result	size
derivativedivides	\({\mathrm e}^{x} \arcsin \left ({\mathrm e}^{x}\right )+\sqrt {1-{\mathrm e}^{2 x}}\)	\(18\)
default	\({\mathrm e}^{x} \arcsin \left ({\mathrm e}^{x}\right )+\sqrt {1-{\mathrm e}^{2 x}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*arcsin(exp(x)),x,method=_RETURNVERBOSE)

[Out]

exp(x)*arcsin(exp(x))+(-exp(x)^2+1)^(1/2)

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Maxima [A]
time = 0.48, size = 17, normalized size = 0.77 \begin {gather*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(e^x)*e^x + sqrt(-e^(2*x) + 1)

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Fricas [A]
time = 2.74, size = 17, normalized size = 0.77 \begin {gather*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(e^x)*e^x + sqrt(-e^(2*x) + 1)

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Sympy [A]
time = 0.21, size = 17, normalized size = 0.77 \begin {gather*} \sqrt {1 - e^{2 x}} + e^{x} \operatorname {asin}{\left (e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(1 - exp(2*x)) + exp(x)*asin(exp(x))

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Giac [A]
time = 0.39, size = 17, normalized size = 0.77 \begin {gather*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(e^x)*e^x + sqrt(-e^(2*x) + 1)

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Mupad [B]
time = 0.34, size = 17, normalized size = 0.77 \begin {gather*} \sqrt {1-{\mathrm {e}}^{2\,x}}+\mathrm {asin}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - exp(2*x))^(1/2) + asin(exp(x))*exp(x)

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