Integrand size = 18, antiderivative size = 245 \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3}-\frac {d e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}-\frac {e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3} \]
d^2*Chi(a/b+arcsinh(c*x))*cosh(a/b)/b/c-1/4*e^2*Chi(a/b+arcsinh(c*x))*cosh (a/b)/b/c^3+1/4*e^2*Chi(3*a/b+3*arcsinh(c*x))*cosh(3*a/b)/b/c^3+d*e*cosh(2 *a/b)*Shi(2*a/b+2*arcsinh(c*x))/b/c^2-d^2*Shi(a/b+arcsinh(c*x))*sinh(a/b)/ b/c+1/4*e^2*Shi(a/b+arcsinh(c*x))*sinh(a/b)/b/c^3-d*e*Chi(2*a/b+2*arcsinh( c*x))*sinh(2*a/b)/b/c^2-1/4*e^2*Shi(3*a/b+3*arcsinh(c*x))*sinh(3*a/b)/b/c^ 3
Time = 0.35 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {\left (4 c^2 d^2-e^2\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 c d e \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-4 c^2 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+4 c d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{4 b c^3} \]
((4*c^2*d^2 - e^2)*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + e^2*Cosh[( 3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 4*c*d*e*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] - 4*c^2*d^2*Sinh[a/b]*SinhIntegral[a/b + A rcSinh[c*x]] + e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 4*c*d*e*Co sh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - e^2*Sinh[(3*a)/b]*SinhI ntegral[3*(a/b + ArcSinh[c*x])])/(4*b*c^3)
Time = 0.93 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx\) |
| \(\Big \downarrow \) 6245 |
| \(\displaystyle \frac {\int \frac {(c d+c e x)^2 \sqrt {c^2 x^2+1}}{a+b \text {arcsinh}(c x)}d\text {arcsinh}(c x)}{c^3}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {c^2 \sqrt {c^2 x^2+1} d^2}{a+b \text {arcsinh}(c x)}+\frac {c e \sinh (2 \text {arcsinh}(c x)) d}{a+b \text {arcsinh}(c x)}+\frac {c^2 e^2 x^2 \sqrt {c^2 x^2+1}}{a+b \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)}{c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c^2 d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b}-\frac {c^2 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b}-\frac {c d e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b}+\frac {c d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b}-\frac {e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b}}{c^3}\) |
((c^2*d^2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/b - (e^2*Cosh[a/b]*C oshIntegral[a/b + ArcSinh[c*x]])/(4*b) + (e^2*Cosh[(3*a)/b]*CoshIntegral[( 3*a)/b + 3*ArcSinh[c*x]])/(4*b) - (c*d*e*CoshIntegral[(2*a)/b + 2*ArcSinh[ c*x]]*Sinh[(2*a)/b])/b - (c^2*d^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x ]])/b + (e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(4*b) + (c*d*e*Co sh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]])/b - (e^2*Sinh[(3*a)/b] *SinhIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(4*b))/c^3
3.1.20.3.1 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 = 1.00 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {-\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {d e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c b}-\frac {d e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c b}}{c}\) | \(254\) |
| default | \(\frac {-\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {d e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c b}-\frac {d e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c b}}{c}\) | \(254\) |
1/c*(-1/8/c^2*e^2/b*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)-1/8/c^2*e^2/b*ex p(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)-1/2/b*exp(a/b)*Ei(1,arcsinh(c*x)+a/b )*d^2+1/8/c^2/b*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)*e^2-1/2/b*exp(-a/b)*Ei(1,- arcsinh(c*x)-a/b)*d^2+1/8/c^2/b*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*e^2+1/2/ c*d*e/b*exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)-1/2/c*d*e/b*exp(-2*a/b)*Ei(1 ,-2*arcsinh(c*x)-2*a/b))
\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\left (d + e x\right )^{2}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]