Integrand size = 18, antiderivative size = 247 \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x^2 \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {d \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^3}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3} \]
d*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c-1/4*e*cosh(a/b)*Shi((a+b*arcsi nh(c*x))/b)/b^2/c^3+3/4*e*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b^2/c^3- d*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c+1/4*e*Chi((a+b*arcsinh(c*x))/b )*sinh(a/b)/b^2/c^3-3/4*e*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^3- d*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))-e*x^2*(c^2*x^2+1)^(1/2)/b/c/(a+ b*arcsinh(c*x))
Time = 0.91 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.77 \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {4 b c^2 d \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {4 b c^2 e x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\left (4 c^2 d-e\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 e \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-4 c^2 d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{4 b^2 c^3} \]
-1/4*((4*b*c^2*d*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*b*c^2*e*x^2* Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*c^2*d - e)*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*e*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[( 3*a)/b] - 4*c^2*d*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e*Cosh[a/b] *SinhIntegral[a/b + ArcSinh[c*x]] - 3*e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^3)
Time = 0.67 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6208, 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.
| \(\displaystyle \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6208 |
| \(\displaystyle \int \left (\frac {d}{(a+b \text {arcsinh}(c x))^2}+\frac {e x^2}{(a+b \text {arcsinh}(c x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}\) |
-((d*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (e*x^2*Sqrt[1 + c^2* x^2])/(b*c*(a + b*ArcSinh[c*x])) - (d*CoshIntegral[(a + b*ArcSinh[c*x])/b] *Sinh[a/b])/(b^2*c) + (e*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/( 4*b^2*c^3) - (3*e*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/ (4*b^2*c^3) + (d*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (e*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^3) + (3*e*Cos h[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3)
3.7.25.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Time = 0.65 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.77
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (4 c^{3} x^{3}-4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+3 c x -\sqrt {c^{2} x^{2}+1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}+3 c x +4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b^{2}}-\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b^{2}}}{c}\) | \(438\) |
| default | \(\frac {\frac {\left (4 c^{3} x^{3}-4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+3 c x -\sqrt {c^{2} x^{2}+1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}+3 c x +4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b^{2}}-\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b^{2}}}{c}\) | \(438\) |
1/c*(1/8*(4*c^3*x^3-4*c^2*x^2*(c^2*x^2+1)^(1/2)+3*c*x-(c^2*x^2+1)^(1/2))*e /c^2/b/(a+b*arcsinh(c*x))+3/8*e/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a /b)-1/8*e/c^2/b*(4*c^3*x^3+3*c*x+4*c^2*x^2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1)^( 1/2))/(a+b*arcsinh(c*x))-3/8*e/c^2/b^2*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3* a/b)+1/2*(-(c^2*x^2+1)^(1/2)+c*x)*d/b/(a+b*arcsinh(c*x))-1/8*(-(c^2*x^2+1) ^(1/2)+c*x)*e/c^2/b/(a+b*arcsinh(c*x))+1/2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+ a/b)*d-1/8/c^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)*e-1/2/b*(c*x+(c^2*x^2+1 )^(1/2))/(a+b*arcsinh(c*x))*d+1/8/c^2/b*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsi nh(c*x))*e-1/2/b^2*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*d+1/8/c^2/b^2*exp(-a/ b)*Ei(1,-arcsinh(c*x)-a/b)*e)
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^3*e*x^5 + (c^3*d + c*e)*x^3 + c*d*x + (c^2*e*x^4 + (c^2*d + e)*x^2 + 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((3*c^5*e*x^6 + (c^5*d + 6*c^3*e)*x^4 + (2*c^3*d + 3*c* e)*x^2 + (3*c^3*e*x^4 + (c^3*d + c*e)*x^2 - c*d)*(c^2*x^2 + 1) + c*d + (6* c^4*e*x^5 + (2*c^4*d + 7*c^2*e)*x^3 + (c^2*d + 2*e)*x)*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)
\[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {d+e x^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]