Integrand size = 16, antiderivative size = 180 \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2} \]
e*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/b^2/c^2+d*cosh(a/b)*Shi((a+b*arc sinh(c*x))/b)/b^2/c-d*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c-e*Shi(2*(a +b*arcsinh(c*x))/b)*sinh(2*a/b)/b^2/c^2-d*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsi nh(c*x))-e*x*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))
Time = 0.89 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83 \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {b c d \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {b c e x \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+c d \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-c d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{b^2 c^2} \]
-(((b*c*d*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (b*c*e*x*Sqrt[1 + c^2* x^2])/(a + b*ArcSinh[c*x]) - e*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh [c*x])] + c*d*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - c*d*Cosh[a/b]*S inhIntegral[a/b + ArcSinh[c*x]] + e*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + Ar cSinh[c*x])])/(b^2*c^2))
Time = 0.53 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6244, 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.
| \(\displaystyle \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6244 |
| \(\displaystyle \int \left (\frac {d}{(a+b \text {arcsinh}(c x))^2}+\frac {e x}{(a+b \text {arcsinh}(c x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}\) |
-((d*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (e*x*Sqrt[1 + c^2*x^ 2])/(b*c*(a + b*ArcSinh[c*x])) + (e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*A rcSinh[c*x]))/b])/(b^2*c^2) - (d*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh [a/b])/(b^2*c) + (d*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c ) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2)
3.1.26.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
Time = 0.56 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.51
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) e}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\) | \(272\) |
| default | \(\frac {\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) e}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\) | \(272\) |
1/c*(1/2*(-(c^2*x^2+1)^(1/2)+c*x)*d/b/(a+b*arcsinh(c*x))+1/2/b^2*exp(a/b)* Ei(1,arcsinh(c*x)+a/b)*d-1/2/b*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))* d-1/2/b^2*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*d+1/4*(-2*c*x*(c^2*x^2+1)^(1/2 )+2*c^2*x^2+1)*e/c/b/(a+b*arcsinh(c*x))-1/2*e/c/b^2*exp(2*a/b)*Ei(1,2*arcs inh(c*x)+2*a/b)-1/4*e/c/b*(2*c^2*x^2+1+2*c*x*(c^2*x^2+1)^(1/2))/(a+b*arcsi nh(c*x))-1/2*e/c/b^2*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b))
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {d + e x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^3*e*x^4 + c^3*d*x^3 + c*e*x^2 + c*d*x + (c^2*e*x^3 + c^2*d*x^2 + e*x + d)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x ^2 + 1))) + integrate((2*c^5*e*x^5 + c^5*d*x^4 + 4*c^3*e*x^3 + 2*c^3*d*x^2 + 2*c*e*x + (2*c^3*e*x^3 + c^3*d*x^2 - c*d)*(c^2*x^2 + 1) + c*d + (4*c^4* e*x^4 + 2*c^4*d*x^3 + 4*c^2*e*x^2 + c^2*d*x + e)*sqrt(c^2*x^2 + 1))/(a*b*c ^5*x^4 + (c^2*x^2 + 1)*a*b*c^3*x^2 + 2*a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2*c^3*x^2 + 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 + b^2 *c^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^3 + a*b*c^2*x)*sqrt(c^2*x^2 + 1)), x)
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {d+e\,x}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]