Integrand size = 27, antiderivative size = 268 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3}-\frac {5 \log (a+b \text {arcsinh}(c x))}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3} \]
-1/32*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/b/c^3+1/32*Chi(4*(a+b*arcsin h(c*x))/b)*cosh(4*a/b)/b/c^3+1/32*Chi(6*(a+b*arcsinh(c*x))/b)*cosh(6*a/b)/ b/c^3+1/128*Chi(8*(a+b*arcsinh(c*x))/b)*cosh(8*a/b)/b/c^3-5/128*ln(a+b*arc sinh(c*x))/b/c^3+1/32*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b/c^3-1/32*S hi(4*(a+b*arcsinh(c*x))/b)*sinh(4*a/b)/b/c^3-1/32*Shi(6*(a+b*arcsinh(c*x)) /b)*sinh(6*a/b)/b/c^3-1/128*Shi(8*(a+b*arcsinh(c*x))/b)*sinh(8*a/b)/b/c^3
Time = 0.73 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+4 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+4 \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 \log (a+b \text {arcsinh}(c x))+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{128 b c^3} \]
(-4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + 4*Cosh[(4*a)/b]*C oshIntegral[4*(a/b + ArcSinh[c*x])] + 4*Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcSinh[c*x])] + Cosh[(8*a)/b]*CoshIntegral[8*(a/b + ArcSinh[c*x])] - 5* Log[a + b*ArcSinh[c*x]] + 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c* x])] - 4*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 4*Sinh[(6*a) /b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - Sinh[(8*a)/b]*SinhIntegral[8*(a /b + ArcSinh[c*x])])/(128*b*c^3)
Time = 0.70 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6234, 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.
| \(\displaystyle \int \frac {x^2 \left (c^2 x^2+1\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {\int \frac {\cosh ^6\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^3}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\int \left (\frac {\cosh \left (\frac {8 a}{b}-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}-\frac {5}{128 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{128} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{128} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{128} \log (a+b \text {arcsinh}(c x))}{b c^3}\) |
(-1/32*(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]) + (Cosh[(4 *a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/32 + (Cosh[(6*a)/b]*CoshI ntegral[(6*(a + b*ArcSinh[c*x]))/b])/32 + (Cosh[(8*a)/b]*CoshIntegral[(8*( a + b*ArcSinh[c*x]))/b])/128 - (5*Log[a + b*ArcSinh[c*x]])/128 + (Sinh[(2* a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/32 - (Sinh[(4*a)/b]*SinhIn tegral[(4*(a + b*ArcSinh[c*x]))/b])/32 - (Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/32 - (Sinh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcSinh [c*x]))/b])/128)/(b*c^3)
3.4.73.3.1 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_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.51 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(-\frac {{\mathrm e}^{\frac {8 a}{b}} \operatorname {Ei}_{1}\left (8 \,\operatorname {arcsinh}\left (c x \right )+\frac {8 a}{b}\right )+4 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {Ei}_{1}\left (6 \,\operatorname {arcsinh}\left (c x \right )+\frac {6 a}{b}\right )+4 \,{\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (c x \right )+\frac {4 a}{b}\right )-4 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )-4 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )+4 \,{\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (c x \right )-\frac {4 a}{b}\right )+4 \,{\mathrm e}^{-\frac {6 a}{b}} \operatorname {Ei}_{1}\left (-6 \,\operatorname {arcsinh}\left (c x \right )-\frac {6 a}{b}\right )+{\mathrm e}^{-\frac {8 a}{b}} \operatorname {Ei}_{1}\left (-8 \,\operatorname {arcsinh}\left (c x \right )-\frac {8 a}{b}\right )+10 \ln \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} b}\) | \(211\) |
-1/256*(exp(8*a/b)*Ei(1,8*arcsinh(c*x)+8*a/b)+4*exp(6*a/b)*Ei(1,6*arcsinh( c*x)+6*a/b)+4*exp(4*a/b)*Ei(1,4*arcsinh(c*x)+4*a/b)-4*exp(2*a/b)*Ei(1,2*ar csinh(c*x)+2*a/b)-4*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)+4*exp(-4*a/b)* Ei(1,-4*arcsinh(c*x)-4*a/b)+4*exp(-6*a/b)*Ei(1,-6*arcsinh(c*x)-6*a/b)+exp( -8*a/b)*Ei(1,-8*arcsinh(c*x)-8*a/b)+10*ln(a+b*arcsinh(c*x)))/c^3/b
\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{5/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]