Integrand size = 18, antiderivative size = 359 \[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {2 d e x \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {e^2 x^2 \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}+\frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^3}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3} \]
2*d*e*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/b^2/c^2+d^2*cosh(a/b)*Shi((a +b*arcsinh(c*x))/b)/b^2/c-1/4*e^2*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/ c^3+3/4*e^2*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b^2/c^3-d^2*Chi((a+b*a rcsinh(c*x))/b)*sinh(a/b)/b^2/c+1/4*e^2*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b )/b^2/c^3-2*d*e*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b^2/c^2-3/4*e^2*Ch i(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^3-d^2*(c^2*x^2+1)^(1/2)/b/c/(a +b*arcsinh(c*x))-2*d*e*x*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))-e^2*x^2* (c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))
Time = 1.66 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {4 b c^2 d^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {8 b c^2 d e x \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {4 b c^2 e^2 x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-8 c d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\left (4 c^2 d^2-e^2\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 e^2 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-4 c^2 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+8 c d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{4 b^2 c^3} \]
-1/4*((4*b*c^2*d^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (8*b*c^2*d*e* x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*b*c^2*e^2*x^2*Sqrt[1 + c^2* x^2])/(a + b*ArcSinh[c*x]) - 8*c*d*e*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + A rcSinh[c*x])] + (4*c^2*d^2 - e^2)*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/ b] + 3*e^2*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 4*c^2*d^2* Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e^2*Cosh[a/b]*SinhIntegral[a/ b + ArcSinh[c*x]] + 8*c*d*e*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c* x])] - 3*e^2*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^3)
Time = 0.85 (sec) , antiderivative size = 359, 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 = {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.
| \(\displaystyle \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6244 |
| \(\displaystyle \int \left (\frac {d^2}{(a+b \text {arcsinh}(c x))^2}+\frac {2 d e x}{(a+b \text {arcsinh}(c x))^2}+\frac {e^2 x^2}{(a+b \text {arcsinh}(c x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3}+\frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {2 d e x \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e^2 x^2 \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}\) |
-((d^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (2*d*e*x*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (e^2*x^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) + (2*d*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[ c*x]))/b])/(b^2*c^2) - (d^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b] )/(b^2*c) + (e^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b^2*c^ 3) - (3*e^2*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2 *c^3) + (d^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (e^ 2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^3) - (2*d*e*Sin h[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2) + (3*e^2*Co sh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3)
3.1.25.3.1 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 = 0.97 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.72
| 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^{2}}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {3 e^{2} {\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^{2} \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 b \,c^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d^{2}}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) e^{2}}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d^{2} \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {e^{2} \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) d e}{2 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{c \,b^{2}}-\frac {e d \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{2 b c \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{b^{2} c}}{c}\) | \(616\) |
| 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^{2}}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {3 e^{2} {\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^{2} \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 b \,c^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d^{2}}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) e^{2}}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d^{2} \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {e^{2} \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) d e}{2 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{c \,b^{2}}-\frac {e d \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{2 b c \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{b^{2} c}}{c}\) | \(616\) |
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 ^2/c^2/b/(a+b*arcsinh(c*x))+3/8*e^2/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x) +3*a/b)-1/8/b*e^2/c^2*(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/b^2*e^2/c^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^2/b/(a+b*arcsinh(c*x))+1/2*d^2 /b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/8*(-(c^2*x^2+1)^(1/2)+c*x)*e^2/c^2/ b/(a+b*arcsinh(c*x))-1/8/c^2*e^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/2/b *d^2*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))-1/2/b^2*d^2*exp(-a/b)*Ei(1 ,-arcsinh(c*x)-a/b)+1/8/c^2/b*e^2*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x ))+1/8/c^2/b^2*e^2*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)+1/2*(-2*c*x*(c^2*x^2+ 1)^(1/2)+2*c^2*x^2+1)*d*e/c/b/(a+b*arcsinh(c*x))-e*d/c/b^2*exp(2*a/b)*Ei(1 ,2*arcsinh(c*x)+2*a/b)-1/2/b*e*d/c*(2*c^2*x^2+1+2*c*x*(c^2*x^2+1)^(1/2))/( a+b*arcsinh(c*x))-1/b^2*e*d/c*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b))
\[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^3*e^2*x^5 + 2*c^3*d*e*x^4 + 2*c*d*e*x^2 + c*d^2*x + (c^3*d^2 + c*e^2)* x^3 + (c^2*e^2*x^4 + 2*c^2*d*e*x^3 + 2*d*e*x + (c^2*d^2 + e^2)*x^2 + d^2)* 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^2*x^6 + 4*c^5*d*e*x^5 + 8*c^3*d*e*x^3 + (c^5*d^ 2 + 6*c^3*e^2)*x^4 + 4*c*d*e*x + c*d^2 + (2*c^3*d^2 + 3*c*e^2)*x^2 + (3*c^ 3*e^2*x^4 + 4*c^3*d*e*x^3 - c*d^2 + (c^3*d^2 + c*e^2)*x^2)*(c^2*x^2 + 1) + (6*c^4*e^2*x^5 + 8*c^4*d*e*x^4 + 8*c^2*d*e*x^2 + (2*c^4*d^2 + 7*c^2*e^2)* x^3 + 2*d*e + (c^2*d^2 + 2*e^2)*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)*sq rt(c^2*x^2 + 1)), x)
\[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^2}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]