\(\int \text {arcsinh}(c e^{a+b x}) \, dx\) [283]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \text {arcsinh}\left (c e^{a+b x}\right ) \, dx=\frac {-\text {arcsinh}\left (c e^{a+b x}\right ) \left (\text {arcsinh}\left (c e^{a+b x}\right )-2 \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )}{2 b} \] Input:

Integrate[ArcSinh[c*E^(a + b*x)],x]
 

Output:

(-(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)
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2720, 6190, 3042, 26, 4199, 25, 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.

\(\displaystyle \int \text {arcsinh}\left (c e^{a+b x}\right ) \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int e^{-a-b x} \text {arcsinh}\left (c e^{a+b x}\right )de^{a+b x}}{b}\)

\(\Big \downarrow \) 6190

\(\displaystyle \frac {\int \frac {e^{-a-b x} \sqrt {e^{2 a+2 b x} c^2+1} \text {arcsinh}\left (c e^{a+b x}\right )}{c}d\text {arcsinh}\left (c e^{a+b x}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i \text {arcsinh}\left (c e^{a+b x}\right ) \tan \left (i \text {arcsinh}\left (c e^{a+b x}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (c e^{a+b x}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \text {arcsinh}\left (c e^{a+b x}\right ) \tan \left (i \text {arcsinh}\left (c e^{a+b x}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (c e^{a+b x}\right )}{b}\)

\(\Big \downarrow \) 4199

\(\displaystyle -\frac {i \left (2 i \int -\frac {e^{a+b x+2 \text {arcsinh}\left (c e^{a+b x}\right )}}{1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}}d\text {arcsinh}\left (c e^{a+b x}\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i \left (-2 i \int \frac {e^{a+b x+2 \text {arcsinh}\left (c e^{a+b x}\right )}}{1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}}d\text {arcsinh}\left (c e^{a+b x}\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )d\text {arcsinh}\left (c e^{a+b x}\right )-\frac {1}{2} \text {arcsinh}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {i \left (-2 i \left (\frac {1}{4} \int e^{-a-b x} \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )de^{2 \text {arcsinh}\left (c e^{a+b x}\right )}-\frac {1}{2} \text {arcsinh}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (c e^{a+b x}\right )}\right )\right )-\frac {1}{2} i e^{2 a+2 b x}\right )}{b}\)

Input:

Int[ArcSinh[c*E^(a + b*x)],x]
 

Output:

((-I)*((-1/2*I)*E^(2*a + 2*b*x) - (2*I)*(-1/2*(ArcSinh[c*E^(a + b*x)]*Log[ 
1 - E^(2*ArcSinh[c*E^(a + b*x)])]) - PolyLog[2, E^(2*ArcSinh[c*E^(a + b*x) 
])]/4)))/b
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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]
 

rule 2715

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]
 

rule 2720

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]]]
 

rule 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

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] + Simp 
[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] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.01

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-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*log(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]

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

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

Output:

integrate(arcsinh(c*e^(b*x + a)), x)
 

Mupad [F(-1)]

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

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

Output:

int(asinh(c*exp(b*x)*exp(a)), x)
 

Reduce [F]

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

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

Output:

int(asinh(e**(a + b*x)*c),x)