\(\int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx\) [21]

Optimal result

Integrand size = 16, antiderivative size = 116 \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b c^2} \] Output:

d*cosh(a/b)*Chi(a/b+arcsinh(c*x))/b/c-1/2*e*Chi(2*a/b+2*arcsinh(c*x))*sinh 
(2*a/b)/b/c^2-d*sinh(a/b)*Shi(a/b+arcsinh(c*x))/b/c+1/2*e*cosh(2*a/b)*Shi( 
2*a/b+2*arcsinh(c*x))/b/c^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\frac {2 c d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-e \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{2 b c^2} \] Input:

Integrate[(d + e*x)/(a + b*ArcSinh[c*x]),x]
 

Output:

(2*c*d*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - e*CoshIntegral[2*(a/b 
+ ArcSinh[c*x])]*Sinh[(2*a)/b] - 2*c*d*Sinh[a/b]*SinhIntegral[a/b + ArcSin 
h[c*x]] + e*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(2*b*c^2)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6245, 7293, 2009}

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.

\(\displaystyle \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx\)

\(\Big \downarrow \) 6245

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {c d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b}-\frac {c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b}}{c^2}\)

Input:

Int[(d + e*x)/(a + b*ArcSinh[c*x]),x]
 

Output:

((c*d*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/b - (e*CoshIntegral[(2*a 
)/b + 2*ArcSinh[c*x]]*Sinh[(2*a)/b])/(2*b) - (c*d*Sinh[a/b]*SinhIntegral[a 
/b + ArcSinh[c*x]])/b + (e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[ 
c*x]])/(2*b))/c^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6245

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[1/c^(m + 1)   Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[ 
x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 
0]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 3.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {-\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 b}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{2 b}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )}{4 c b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )}{4 c b}}{c}\)	\(120\)
default	\(\frac {-\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 b}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{2 b}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )}{4 c b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )}{4 c b}}{c}\)	\(120\)

Input:

int((e*x+d)/(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

1/c*(-1/2*d/b*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)-1/2*d/b*exp(-a/b)*Ei(1,-arcs 
inh(x*c)-a/b)+1/4*e/c/b*exp(2*a/b)*Ei(1,2*arcsinh(x*c)+2*a/b)-1/4*e/c/b*ex 
p(-2*a/b)*Ei(1,-2*arcsinh(x*c)-2*a/b))
 

Fricas [F]

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

integrate((e*x+d)/(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

integral((e*x + d)/(b*arcsinh(c*x) + a), x)
 

Sympy [F]

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

integrate((e*x+d)/(a+b*asinh(c*x)),x)
 

Output:

Integral((d + e*x)/(a + b*asinh(c*x)), x)
 

Maxima [F]

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

integrate((e*x+d)/(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x+d)/(a+b*arcsinh(c*x)),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\left (\int \frac {x}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) e +\left (\int \frac {1}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) d \] Input:

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

Output:

int(x/(asinh(c*x)*b + a),x)*e + int(1/(asinh(c*x)*b + a),x)*d