Integrand size = 27, antiderivative size = 141 \[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^4}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c^5}-\frac {\text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^5}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^5}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b^2 c^5} \]
-x^4/b/c/(a+b*arcsinh(c*x))-cosh(2*a/b)*Shi(2*(a+b*arcsinh(c*x))/b)/b^2/c^ 5+1/2*cosh(4*a/b)*Shi(4*(a+b*arcsinh(c*x))/b)/b^2/c^5+Chi(2*(a+b*arcsinh(c *x))/b)*sinh(2*a/b)/b^2/c^5-1/2*Chi(4*(a+b*arcsinh(c*x))/b)*sinh(4*a/b)/b^ 2/c^5
Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\frac {-\frac {2 b c^4 x^4}{a+b \text {arcsinh}(c x)}+2 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-\text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{2 b^2 c^5} \]
((-2*b*c^4*x^4)/(a + b*ArcSinh[c*x]) + 2*CoshIntegral[2*(a/b + ArcSinh[c*x ])]*Sinh[(2*a)/b] - CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] - 2 *Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + Cosh[(4*a)/b]*SinhIn tegral[4*(a/b + ArcSinh[c*x])])/(2*b^2*c^5)
Time = 0.63 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6233, 6195, 25, 5971, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {4 \int \frac {x^3}{a+b \text {arcsinh}(c x)}dx}{b c}-\frac {x^4}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {4 \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {4 \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 (a+b \text {arcsinh}(c x))}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{4 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \text {arcsinh}(c x))}\) |
-(x^4/(b*c*(a + b*ArcSinh[c*x]))) + (4*((CoshIntegral[(2*(a + b*ArcSinh[c* x]))/b]*Sinh[(2*a)/b])/4 - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[ (4*a)/b])/8 - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/4 + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8))/(b^2*c^5)
3.5.36.3.1 Defintions of rubi rules used
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Time = 0.33 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.72
| method | result | size |
| default | \(-\frac {4 b \,c^{4} x^{4}+{\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (c x \right )-\frac {4 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )-{\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (c x \right )+\frac {4 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )+2 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )-2 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )+{\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (c x \right )-\frac {4 a}{b}\right ) a -{\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (c x \right )+\frac {4 a}{b}\right ) a +2 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right ) a -2 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right ) a}{4 c^{5} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\) | \(243\) |
-1/4*(4*b*c^4*x^4+exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)*b*arcsinh(c*x)-e xp(4*a/b)*Ei(1,4*arcsinh(c*x)+4*a/b)*b*arcsinh(c*x)+2*exp(2*a/b)*Ei(1,2*ar csinh(c*x)+2*a/b)*b*arcsinh(c*x)-2*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b) *b*arcsinh(c*x)+exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)*a-exp(4*a/b)*Ei(1, 4*arcsinh(c*x)+4*a/b)*a+2*exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)*a-2*exp(-2 *a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)*a)/c^5/b^2/(a+b*arcsinh(c*x))
\[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{4}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
integral(sqrt(c^2*x^2 + 1)*x^4/(a^2*c^2*x^2 + (b^2*c^2*x^2 + b^2)*arcsinh( c*x)^2 + a^2 + 2*(a*b*c^2*x^2 + a*b)*arcsinh(c*x)), x)
\[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{4}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]
\[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{4}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^3*x^7 + c*x^5 + (c^2*x^6 + x^4)*sqrt(c^2*x^2 + 1))/((c^2*x^2 + 1)*a*b* c^2*x + ((c^2*x^2 + 1)*b^2*c^2*x + (b^2*c^3*x^2 + b^2*c)*sqrt(c^2*x^2 + 1) )*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)) + integrate((4*c^5*x^8 + 9*c^3*x^6 + 5*c*x^4 + (4*c^3*x^6 + 3*c*x^4)*(c^2* x^2 + 1) + 4*(2*c^4*x^7 + 3*c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1))/((c^2*x^2 + 1)^(3/2)*a*b*c^3*x^2 + 2*(a*b*c^4*x^3 + a*b*c^2*x)*(c^2*x^2 + 1) + ((c^2*x ^2 + 1)^(3/2)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 + b^2*c^2*x)*(c^2*x^2 + 1) + (b ^2*c^5*x^4 + 2*b^2*c^3*x^2 + b^2*c)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2* x^2 + 1)) + (a*b*c^5*x^4 + 2*a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)), x)
\[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{4}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^4}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]