\(\int \text {arcsinh}(c e^{a+b x}) \, dx\) [348]

Optimal result

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

[Out]

-1/2*arcsinh(c*exp(b*x+a))^2/b+arcsinh(c*exp(b*x+a))*ln(1-(c*exp(b*x+a)+(1+c^2*exp(b*x+a)^2)^(1/2))^2)/b+1/2*p
olylog(2,(c*exp(b*x+a)+(1+c^2*exp(b*x+a)^2)^(1/2))^2)/b

Rubi [A] (verified)

Time = 0.06 (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, 5775, 3797, 2221, 2317, 2438} \[ \int \text {arcsinh}\left (c e^{a+b x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\text {arcsinh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arcsinh}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )}{b} \]

[In]

Int[ArcSinh[c*E^(a + b*x)],x]

[Out]

-1/2*ArcSinh[c*E^(a + b*x)]^2/b + (ArcSinh[c*E^(a + b*x)]*Log[1 - E^(2*ArcSinh[c*E^(a + b*x)])])/b + PolyLog[2
, E^(2*ArcSinh[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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

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

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \text {arcsinh}\left (c e^{a+b x}\right ) \, dx=\frac {-\text {arcsinh}\left (c e^{a+b x}\right ) \left (\text {arcsinh}\left (c e^{a+b x}\right )-2 \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )}{2 b} \]

[In]

Integrate[ArcSinh[c*E^(a + b*x)],x]

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arcsinh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arcsinh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+c \,{\mathrm e}^{b x +a}+\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )+\operatorname {polylog}\left (2, -c \,{\mathrm e}^{b x +a}-\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )+\operatorname {arcsinh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1-c \,{\mathrm e}^{b x +a}-\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )+\operatorname {polylog}\left (2, c \,{\mathrm e}^{b x +a}+\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}\)	\(153\)
default	\(\frac {-\frac {\operatorname {arcsinh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arcsinh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+c \,{\mathrm e}^{b x +a}+\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )+\operatorname {polylog}\left (2, -c \,{\mathrm e}^{b x +a}-\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )+\operatorname {arcsinh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1-c \,{\mathrm e}^{b x +a}-\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )+\operatorname {polylog}\left (2, c \,{\mathrm e}^{b x +a}+\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}\)	\(153\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(arcsinh(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]

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

[In]

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

[Out]

Integral(asinh(c*exp(a + b*x)), x)

Maxima [F]

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

[In]

integrate(arcsinh(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)^(3/2)), x) + x*l
og(c*e^(b*x + a) + sqrt(c^2*e^(2*b*x + 2*a) + 1)) - 1/4*(2*b*x*log(c^2*e^(2*b*x + 2*a) + 1) + dilog(-c^2*e^(2*
b*x + 2*a)))/b

Giac [F]

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

[In]

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

[Out]

integrate(arcsinh(c*e^(b*x + a)), x)

Mupad [F(-1)]

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

[In]

int(asinh(c*exp(a + b*x)),x)

[Out]

int(asinh(c*exp(b*x)*exp(a)), x)