Integrand size = 10, antiderivative size = 76 \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=-\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b} \]
-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
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \text {arccosh}\left (c e^{a+b x}\right ) \, dx \]
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {2720, 6297, 3042, 26, 4201, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int e^{-a-b x} \text {arccosh}\left (c e^{a+b x}\right )de^{a+b x}}{b}\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle \frac {\int \frac {e^{-a-b x} \sqrt {\frac {c e^{a+b x}-1}{e^{a+b x} c+1}} \left (e^{a+b x} c+1\right ) \text {arccosh}\left (c e^{a+b x}\right )}{c}d\text {arccosh}\left (c e^{a+b x}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -i \text {arccosh}\left (c e^{a+b x}\right ) \tan \left (i \text {arccosh}\left (c e^{a+b x}\right )\right )d\text {arccosh}\left (c e^{a+b x}\right )}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i \int \text {arccosh}\left (c e^{a+b x}\right ) \tan \left (i \text {arccosh}\left (c e^{a+b x}\right )\right )d\text {arccosh}\left (c e^{a+b x}\right )}{b}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {i \left (2 i \int \frac {e^{a+b x+2 \text {arccosh}\left (c e^{a+b x}\right )}}{1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}}d\text {arccosh}\left (c e^{a+b x}\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (c e^{a+b x}\right ) \log \left (e^{2 \text {arccosh}\left (c e^{a+b x}\right )}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )d\text {arccosh}\left (c e^{a+b x}\right )\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (c e^{a+b x}\right ) \log \left (e^{2 \text {arccosh}\left (c e^{a+b x}\right )}+1\right )-\frac {1}{4} \int e^{-a-b x} \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )de^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )+\frac {1}{2} \text {arccosh}\left (c e^{a+b x}\right ) \log \left (e^{2 \text {arccosh}\left (c e^{a+b x}\right )}+1\right )\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\) |
((-I)*((-1/2*I)*E^(2*a + 2*b*x) + (2*I)*((ArcCosh[c*E^(a + b*x)]*Log[1 + E ^(2*ArcCosh[c*E^(a + b*x)])])/2 + PolyLog[2, -E^(2*ArcCosh[c*E^(a + b*x)]) ]/4)))/b
3.3.74.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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] - Si mp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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]
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] + Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[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]
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 | \(\frac {-\frac {\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{2}}{b}\) | \(110\) |
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))
Exception generated. \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \operatorname {acosh}{\left (c e^{a + b x} \right )}\, dx \]
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right ) \,d x } \]
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) + dilog(c*e^(b*x + a)))/b
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right ) \,d x } \]
Timed out. \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \mathrm {acosh}\left (c\,{\mathrm {e}}^{a+b\,x}\right ) \,d x \]