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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5830, 6874, 3384, 3379, 3382, 5556, 12} \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b c^2}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b c^2}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c} \]

[In]

Int[(d + e*x)/(a + b*ArcSinh[c*x]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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 5556

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 5830

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Dist[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 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) (c d+e \sinh (x))}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^2} \\ & = \frac {\text {Subst}\left (\int \left (\frac {c d \cosh (x)}{a+b x}+\frac {e \cosh (x) \sinh (x)}{a+b x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^2} \\ & = \frac {d \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c}+\frac {e \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^2} \\ & = \frac {e \text {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\text {arcsinh}(c x)\right )}{c^2}+\frac {\left (d \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c}-\frac {\left (d \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c} \\ & = \frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {e \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{2 c^2} \\ & = \frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {\left (e \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{2 c^2}-\frac {\left (e \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{2 c^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\frac {2 c d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-e \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{2 b c^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((d + e*x)/(a + b*asinh(c*x)),x)

[Out]

int((d + e*x)/(a + b*asinh(c*x)), x)