3.274 \(\int \cosh ^{-1}(c e^{a+b x}) \, dx\)

Optimal. Leaf size=76 \[ \frac{\text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac{\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac{\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}+1\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.073347, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {2282, 5660, 3718, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac{\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac{\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}+1\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*E^(a + b*x)],x]

[Out]

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

Rule 2282

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 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

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

Mathematica [F]  time = 0.673708, size = 0, normalized size = 0. \[ \int \cosh ^{-1}\left (c e^{a+b x}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcCosh[c*E^(a + b*x)],x]

[Out]

Integrate[ArcCosh[c*E^(a + b*x)], x]

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Maple [A]  time = 0.02, size = 115, normalized size = 1.5 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (c{{\rm e}^{bx+a}}\right ) \right ) ^{2}}{2\,b}}+{\frac{{\rm arccosh} \left (c{{\rm e}^{bx+a}}\right )}{b}\ln \left ( 1+ \left ( c{{\rm e}^{bx+a}}+\sqrt{c{{\rm e}^{bx+a}}-1}\sqrt{c{{\rm e}^{bx+a}}+1} \right ) ^{2} \right ) }+{\frac{1}{2\,b}{\it polylog} \left ( 2,- \left ( c{{\rm e}^{bx+a}}+\sqrt{c{{\rm e}^{bx+a}}-1}\sqrt{c{{\rm e}^{bx+a}}+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(c*exp(b*x+a)),x)

[Out]

-1/2*arccosh(c*exp(b*x+a))^2/b+arccosh(c*exp(b*x+a))*ln(1+(c*exp(b*x+a)+(c*exp(b*x+a)-1)^(1/2)*(c*exp(b*x+a)+1
)^(1/2))^2)/b+1/2*polylog(2,-(c*exp(b*x+a)+(c*exp(b*x+a)-1)^(1/2)*(c*exp(b*x+a)+1)^(1/2))^2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b c \int \frac{x e^{\left (b x + a\right )}}{c^{3} e^{\left (3 \, b x + 3 \, a\right )} - c e^{\left (b x + a\right )} +{\left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (c e^{\left (b x + a\right )} + 1\right ) + \frac{1}{2} \, \log \left (c e^{\left (b x + a\right )} - 1\right )\right )}}\,{d x} + x \log \left (c e^{\left (b x + a\right )} + \sqrt{c e^{\left (b x + a\right )} + 1} \sqrt{c e^{\left (b x + a\right )} - 1}\right ) - \frac{b x \log \left (c e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-c e^{\left (b x + a\right )}\right )}{2 \, b} - \frac{b x \log \left (-c e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (c e^{\left (b x + a\right )}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*c*integrate(x*e^(b*x + a)/(c^3*e^(3*b*x + 3*a) - c*e^(b*x + a) + (c^2*e^(2*b*x + 2*a) - 1)*e^(1/2*log(c*e^(b
*x + a) + 1) + 1/2*log(c*e^(b*x + a) - 1))), x) + x*log(c*e^(b*x + a) + sqrt(c*e^(b*x + a) + 1)*sqrt(c*e^(b*x
+ a) - 1)) - 1/2*(b*x*log(c*e^(b*x + a) + 1) + dilog(-c*e^(b*x + a)))/b - 1/2*(b*x*log(-c*e^(b*x + a) + 1) + d
ilog(c*e^(b*x + a)))/b

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acosh}{\left (c e^{a + b x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*exp(b*x+a)),x)

[Out]

Integral(acosh(c*exp(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcosh}\left (c e^{\left (b x + a\right )}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*exp(b*x+a)),x, algorithm="giac")

[Out]

integrate(arccosh(c*e^(b*x + a)), x)