\(\int \text {arccosh}(c e^{a+b x}) \, dx\) [274]

Optimal result

Integrand size = 10, antiderivative size = 76 \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=-\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b} \]

[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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2320, 5882, 3799, 2221, 2317, 2438} \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (e^{2 \text {arccosh}\left (c e^{a+b x}\right )}+1\right )}{b} \]

[In]

Int[ArcCosh[c*E^(a + b*x)],x]

[Out]

-1/2*ArcCosh[c*E^(a + b*x)]^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 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 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 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 3799

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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

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

Mathematica [F]

\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \text {arccosh}\left (c e^{a+b x}\right ) \, dx \]

[In]

Integrate[ArcCosh[c*E^(a + b*x)],x]

[Out]

Integrate[ArcCosh[c*E^(a + b*x)], x]

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{2}}{b}\)	\(110\)
default	\(\frac {-\frac {\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{2}}{b}\)	\(110\)

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \operatorname {acosh}{\left (c e^{a + b x} \right )}\, dx \]

[In]

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

[Out]

Integral(acosh(c*exp(a + b*x)), x)

Maxima [F]

\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right ) \,d x } \]

[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

Giac [F]

\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right ) \,d x } \]

[In]

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

[Out]

integrate(arccosh(c*e^(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \mathrm {acosh}\left (c\,{\mathrm {e}}^{a+b\,x}\right ) \,d x \]

[In]

int(acosh(c*exp(a + b*x)),x)

[Out]

int(acosh(c*exp(a + b*x)), x)