3.2.59 \(\int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx\) [159]

3.2.59.1 Optimal result

Integrand size = 21, antiderivative size = 69 \[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=-\frac {e \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b d} \]

output

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

3.2.59.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=-\frac {e \left (\text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c+d x)\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c+d x)\right )\right )}{2 b d} \]

input

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

output

-1/2*(e*(CoshIntegral[(2*a)/b + 2*ArcSinh[c + d*x]]*Sinh[(2*a)/b] - Cosh[( 
2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c + d*x]]))/(b*d)
 

3.2.59.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6274, 27, 6195, 25, 5971, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6195

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

\(\displaystyle \frac {i e \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{2 b d}\)

\(\Big \downarrow \) 3782

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

input

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

output

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

3.2.59.3.1 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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6195

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

rule 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.2.59.4 Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b}}{d}\)	\(66\)
default	\(\frac {\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b}}{d}\)	\(66\)

input

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

output

1/d*(1/4*e/b*exp(2*a/b)*Ei(1,2*arcsinh(d*x+c)+2*a/b)-1/4*e/b*exp(-2*a/b)*E 
i(1,-2*arcsinh(d*x+c)-2*a/b))
 

3.2.59.5 Fricas [F]

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

input

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

output

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

3.2.59.6 Sympy [F]

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

input

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

output

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

3.2.59.7 Maxima [F]

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

input

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

output

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

3.2.59.8 Giac [F]

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

input

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

output

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

3.2.59.9 Mupad [F(-1)]

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

input

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

output

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