Integrand size = 16, antiderivative size = 116 \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b c^2} \]
d*Chi(a/b+arcsinh(c*x))*cosh(a/b)/b/c+1/2*e*cosh(2*a/b)*Shi(2*a/b+2*arcsin h(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*arcs inh(c*x))*sinh(2*a/b)/b/c^2
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\frac {2 c d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-e \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{2 b c^2} \]
(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)
Time = 0.57 (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 |
\(\displaystyle \frac {\frac {c d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b}-\frac {c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{2 b}}{c^2}\) |
((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
3.1.21.3.1 Defintions of rubi rules used
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]
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 | \(\frac {-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{4 c b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{4 c b}}{c}\) | \(120\) |
1/c*(-1/2/b*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)*d-1/2/b*exp(-a/b)*Ei(1,-arcsin h(c*x)-a/b)*d+1/4*e/c/b*exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)-1/4*e/c/b*ex p(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b))
\[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {e x + d}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {d + e x}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {e x + d}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {e x + d}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {d+e x}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {d+e\,x}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]