Integrand size = 20, antiderivative size = 670 \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {3 d^2 e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}-\frac {5 e^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}+\frac {3 d^2 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}-\frac {9 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {9 e^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {3 d e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {5 e^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {e^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}+\frac {3 d^2 e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}-\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}+\frac {5 e^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}-\frac {3 d^2 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}+\frac {9 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {9 e^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {3 d e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {5 e^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {e^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7} \]
d^3*Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/b/c-3/4*d^2*e*Chi((a+b*arcsinh(c*x ))/b)*cosh(a/b)/b/c^3+3/8*d*e^2*Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/b/c^5- 5/64*e^3*Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/b/c^7+3/4*d^2*e*Chi(3*(a+b*ar csinh(c*x))/b)*cosh(3*a/b)/b/c^3-9/16*d*e^2*Chi(3*(a+b*arcsinh(c*x))/b)*co sh(3*a/b)/b/c^5+9/64*e^3*Chi(3*(a+b*arcsinh(c*x))/b)*cosh(3*a/b)/b/c^7+3/1 6*d*e^2*Chi(5*(a+b*arcsinh(c*x))/b)*cosh(5*a/b)/b/c^5-5/64*e^3*Chi(5*(a+b* arcsinh(c*x))/b)*cosh(5*a/b)/b/c^7+1/64*e^3*Chi(7*(a+b*arcsinh(c*x))/b)*co sh(7*a/b)/b/c^7-d^3*Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b/c+3/4*d^2*e*Shi( (a+b*arcsinh(c*x))/b)*sinh(a/b)/b/c^3-3/8*d*e^2*Shi((a+b*arcsinh(c*x))/b)* sinh(a/b)/b/c^5+5/64*e^3*Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b/c^7-3/4*d^2 *e*Shi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b/c^3+9/16*d*e^2*Shi(3*(a+b*arc sinh(c*x))/b)*sinh(3*a/b)/b/c^5-9/64*e^3*Shi(3*(a+b*arcsinh(c*x))/b)*sinh( 3*a/b)/b/c^7-3/16*d*e^2*Shi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b/c^5+5/64 *e^3*Shi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b/c^7-1/64*e^3*Shi(7*(a+b*arc sinh(c*x))/b)*sinh(7*a/b)/b/c^7
Time = 0.75 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\frac {\left (64 c^6 d^3-48 c^4 d^2 e+24 c^2 d e^2-5 e^3\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 e \left (16 c^4 d^2-12 c^2 d e+3 e^2\right ) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+12 c^2 d e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 e^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+e^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-64 c^6 d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+48 c^4 d^2 e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-24 c^2 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+5 e^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-48 c^4 d^2 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+36 c^2 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-9 e^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-12 c^2 d e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 e^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-e^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b c^7} \]
((64*c^6*d^3 - 48*c^4*d^2*e + 24*c^2*d*e^2 - 5*e^3)*Cosh[a/b]*CoshIntegral [a/b + ArcSinh[c*x]] + 3*e*(16*c^4*d^2 - 12*c^2*d*e + 3*e^2)*Cosh[(3*a)/b] *CoshIntegral[3*(a/b + ArcSinh[c*x])] + 12*c^2*d*e^2*Cosh[(5*a)/b]*CoshInt egral[5*(a/b + ArcSinh[c*x])] - 5*e^3*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] + e^3*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 64*c^6*d^3*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 48*c^4*d^2*e*Sinh[ a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 24*c^2*d*e^2*Sinh[a/b]*SinhIntegra l[a/b + ArcSinh[c*x]] + 5*e^3*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 48*c^4*d^2*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 36*c^2* d*e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 9*e^3*Sinh[(3*a )/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 12*c^2*d*e^2*Sinh[(5*a)/b]*Sin hIntegral[5*(a/b + ArcSinh[c*x])] + 5*e^3*Sinh[(5*a)/b]*SinhIntegral[5*(a/ b + ArcSinh[c*x])] - e^3*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x]) ])/(64*b*c^7)
Time = 1.48 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx\) |
| \(\Big \downarrow \) 6208 |
| \(\displaystyle \int \left (\frac {d^3}{a+b \text {arcsinh}(c x)}+\frac {3 d^2 e x^2}{a+b \text {arcsinh}(c x)}+\frac {3 d e^2 x^4}{a+b \text {arcsinh}(c x)}+\frac {e^3 x^6}{a+b \text {arcsinh}(c x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 e^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}+\frac {9 e^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {5 e^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {e^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {5 e^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}-\frac {9 e^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {5 e^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {e^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}-\frac {9 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {3 d e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}+\frac {9 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {3 d e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {3 d^2 e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}+\frac {3 d^2 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}+\frac {3 d^2 e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}-\frac {3 d^2 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}\) |
(d^3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (3*d^2*e*Cosh [a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) + (3*d*e^2*Cosh[a/b] *CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^5) - (5*e^3*Cosh[a/b]*CoshIn tegral[(a + b*ArcSinh[c*x])/b])/(64*b*c^7) + (3*d^2*e*Cosh[(3*a)/b]*CoshIn tegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3) - (9*d*e^2*Cosh[(3*a)/b]*Cos hIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (9*e^3*Cosh[(3*a)/b]*C oshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) + (3*d*e^2*Cosh[(5*a)/ b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (5*e^3*Cosh[(5*a )/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) + (e^3*Cosh[(7*a )/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) - (d^3*Sinh[a/b] *SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) + (3*d^2*e*Sinh[a/b]*SinhInte gral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) - (3*d*e^2*Sinh[a/b]*SinhIntegral[ (a + b*ArcSinh[c*x])/b])/(8*b*c^5) + (5*e^3*Sinh[a/b]*SinhIntegral[(a + b* ArcSinh[c*x])/b])/(64*b*c^7) - (3*d^2*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3) + (9*d*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*( a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (9*e^3*Sinh[(3*a)/b]*SinhIntegral[(3 *(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) - (3*d*e^2*Sinh[(5*a)/b]*SinhIntegra l[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (5*e^3*Sinh[(5*a)/b]*SinhInteg ral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) - (e^3*Sinh[(7*a)/b]*SinhInteg ral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7)
3.7.18.3.1 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 = 1.76 (sec) , antiderivative size = 654, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(\frac {-\frac {e^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {Ei}_{1}\left (7 \,\operatorname {arcsinh}\left (c x \right )+\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {e^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {Ei}_{1}\left (-7 \,\operatorname {arcsinh}\left (c x \right )-\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\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 ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{128 c^{6} b}-\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 ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right )}{128 c^{6} b}}{c}\) | \(654\) |
| default | \(\frac {-\frac {e^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {Ei}_{1}\left (7 \,\operatorname {arcsinh}\left (c x \right )+\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {e^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {Ei}_{1}\left (-7 \,\operatorname {arcsinh}\left (c x \right )-\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\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 ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{128 c^{6} b}-\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 ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right )}{128 c^{6} b}}{c}\) | \(654\) |
1/c*(-1/128/c^6*e^3/b*exp(7*a/b)*Ei(1,7*arcsinh(c*x)+7*a/b)-1/128/c^6*e^3/ b*exp(-7*a/b)*Ei(1,-7*arcsinh(c*x)-7*a/b)-1/2/b*exp(a/b)*Ei(1,arcsinh(c*x) +a/b)*d^3+3/8/c^2/b*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)*d^2*e-3/16/c^4/b*exp(a /b)*Ei(1,arcsinh(c*x)+a/b)*d*e^2+5/128/c^6/b*exp(a/b)*Ei(1,arcsinh(c*x)+a/ b)*e^3-1/2/b*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*d^3+3/8/c^2/b*exp(-a/b)*Ei( 1,-arcsinh(c*x)-a/b)*d^2*e-3/16/c^4/b*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*d* e^2+5/128/c^6/b*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*e^3-3/8/c^2*e/b*exp(3*a/ b)*Ei(1,3*arcsinh(c*x)+3*a/b)*d^2+9/32/c^4*e^2/b*exp(3*a/b)*Ei(1,3*arcsinh (c*x)+3*a/b)*d-9/128/c^6*e^3/b*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)-3/8/c ^2*e/b*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)*d^2+9/32/c^4*e^2/b*exp(-3*a /b)*Ei(1,-3*arcsinh(c*x)-3*a/b)*d-9/128/c^6*e^3/b*exp(-3*a/b)*Ei(1,-3*arcs inh(c*x)-3*a/b)-3/32/c^4*e^2/b*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)*d+5/1 28/c^6*e^3/b*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)-3/32/c^4*e^2/b*exp(-5*a /b)*Ei(1,-5*arcsinh(c*x)-5*a/b)*d+5/128/c^6*e^3/b*exp(-5*a/b)*Ei(1,-5*arcs inh(c*x)-5*a/b))
\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\left (d + e x^{2}\right )^{3}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]