Integrand size = 27, antiderivative size = 204 \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^6} \]
-x^5/b/c/(a+b*arcsinh(c*x))+5/8*Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/b^2/c^ 6-15/16*Chi(3*(a+b*arcsinh(c*x))/b)*cosh(3*a/b)/b^2/c^6+5/16*Chi(5*(a+b*ar csinh(c*x))/b)*cosh(5*a/b)/b^2/c^6-5/8*Shi((a+b*arcsinh(c*x))/b)*sinh(a/b) /b^2/c^6+15/16*Shi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^6-5/16*Shi(5* (a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b^2/c^6
Time = 0.31 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.77 \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \left (2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )\right )}{16 b^2 c^6} \]
-(x^5/(b*c*(a + b*ArcSinh[c*x]))) + (5*(2*Cosh[a/b]*CoshIntegral[a/b + Arc Sinh[c*x]] - 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + Cosh[( 5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 2*Sinh[a/b]*SinhIntegral[a/ b + ArcSinh[c*x]] + 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])]))/(16*b^2*c^6)
Time = 0.68 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6233, 6195, 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.
| \(\displaystyle \int \frac {x^5}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {5 \int \frac {x^4}{a+b \text {arcsinh}(c x)}dx}{b c}-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {5 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^6}-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {5 \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^6}-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \left (\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {3}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {3}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^6}-\frac {x^5}{b c (a+b \text {arcsinh}(c x))}\) |
-(x^5/(b*c*(a + b*ArcSinh[c*x]))) + (5*((Cosh[a/b]*CoshIntegral[(a + b*Arc Sinh[c*x])/b])/8 - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/ b])/16 + (Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/16 - (Si nh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/8 + (3*Sinh[(3*a)/b]*SinhInt egral[(3*(a + b*ArcSinh[c*x]))/b])/16 - (Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/16))/(b^2*c^6)
3.5.35.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(632\) vs. \(2(192)=384\).
Time = 0.27 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.10
| method | result | size |
| default | \(-\frac {16 c^{5} x^{5}-16 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+20 c^{3} x^{3}-12 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+5 c x -\sqrt {c^{2} x^{2}+1}}{32 c^{6} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )}{32 c^{6} b^{2}}+\frac {\frac {5 c^{3} x^{3}}{8}-\frac {5 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{8}+\frac {15 c x}{32}-\frac {5 \sqrt {c^{2} x^{2}+1}}{32}}{c^{6} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {15 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{32 c^{6} b^{2}}-\frac {5 \left (-\sqrt {c^{2} x^{2}+1}+c x \right )}{16 c^{6} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{16 c^{6} b^{2}}-\frac {5 \left (\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a}{b}} b +\operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a}{b}} a +b c x +\sqrt {c^{2} x^{2}+1}\, b \right )}{16 c^{6} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {\frac {5 b \,c^{3} x^{3}}{8}+\frac {5 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}}{8}+\frac {15 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} b}{32}+\frac {15 \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} a}{32}+\frac {15 b c x}{32}+\frac {5 \sqrt {c^{2} x^{2}+1}\, b}{32}}{c^{6} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {16 b \,c^{5} x^{5}+16 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+20 b \,c^{3} x^{3}+12 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+5 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {5 a}{b}} b +5 \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {5 a}{b}} a +5 b c x +\sqrt {c^{2} x^{2}+1}\, b}{32 c^{6} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\) | \(633\) |
-1/32*(16*c^5*x^5-16*c^4*x^4*(c^2*x^2+1)^(1/2)+20*c^3*x^3-12*c^2*x^2*(c^2* x^2+1)^(1/2)+5*c*x-(c^2*x^2+1)^(1/2))/c^6/b/(a+b*arcsinh(c*x))-5/32/c^6/b^ 2*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)+5/32*(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))/c^6/b/(a+b*arcsinh(c*x))+15/32/c^6/b^2* exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)-5/16*(-(c^2*x^2+1)^(1/2)+c*x)/c^6/b/ (a+b*arcsinh(c*x))-5/16/c^6/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-5/16/c^6/b ^2*(arcsinh(c*x)*Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*b+Ei(1,-arcsinh(c*x)-a/ b)*exp(-a/b)*a+b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))+5/32/c^6/b^2* (4*b*c^3*x^3+4*(c^2*x^2+1)^(1/2)*b*c^2*x^2+3*arcsinh(c*x)*Ei(1,-3*arcsinh( c*x)-3*a/b)*exp(-3*a/b)*b+3*Ei(1,-3*arcsinh(c*x)-3*a/b)*exp(-3*a/b)*a+3*b* c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))-1/32/c^6/b^2*(16*b*c^5*x^5+16* (c^2*x^2+1)^(1/2)*b*c^4*x^4+20*b*c^3*x^3+12*(c^2*x^2+1)^(1/2)*b*c^2*x^2+5* arcsinh(c*x)*Ei(1,-5*arcsinh(c*x)-5*a/b)*exp(-5*a/b)*b+5*Ei(1,-5*arcsinh(c *x)-5*a/b)*exp(-5*a/b)*a+5*b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))
\[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{5}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
integral(sqrt(c^2*x^2 + 1)*x^5/(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)
\[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{5}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]
\[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{5}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^3*x^8 + c*x^6 + (c^2*x^7 + x^5)*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((5*c^5*x^9 + 11*c^3*x^7 + 6*c*x^5 + (5*c^3*x^7 + 4*c*x^5)*(c^2 *x^2 + 1) + 5*(2*c^4*x^8 + 3*c^2*x^6 + x^4)*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)
Exception generated. \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^5}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]