Integrand size = 27, antiderivative size = 245 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {3 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{128 b c^4}+\frac {\text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b c^4}-\frac {3 \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{256 b c^4}-\frac {\text {Chi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {9 a}{b}\right )}{256 b c^4}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 b c^4}-\frac {\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^4}+\frac {3 \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b c^4}+\frac {\cosh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b c^4} \]
-3/128*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b/c^4-1/32*cosh(3*a/b)*Shi(3*(a +b*arcsinh(c*x))/b)/b/c^4+3/256*cosh(7*a/b)*Shi(7*(a+b*arcsinh(c*x))/b)/b/ c^4+1/256*cosh(9*a/b)*Shi(9*(a+b*arcsinh(c*x))/b)/b/c^4+3/128*Chi((a+b*arc sinh(c*x))/b)*sinh(a/b)/b/c^4+1/32*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b) /b/c^4-3/256*Chi(7*(a+b*arcsinh(c*x))/b)*sinh(7*a/b)/b/c^4-1/256*Chi(9*(a+ b*arcsinh(c*x))/b)*sinh(9*a/b)/b/c^4
Time = 0.81 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {6 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+8 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )-\text {Chi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {9 a}{b}\right )-6 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-8 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+3 \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{256 b c^4} \]
(6*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 8*CoshIntegral[3*(a/b + Ar cSinh[c*x])]*Sinh[(3*a)/b] - 3*CoshIntegral[7*(a/b + ArcSinh[c*x])]*Sinh[( 7*a)/b] - CoshIntegral[9*(a/b + ArcSinh[c*x])]*Sinh[(9*a)/b] - 6*Cosh[a/b] *SinhIntegral[a/b + ArcSinh[c*x]] - 8*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 3*Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + Co sh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c*x])])/(256*b*c^4)
Time = 0.69 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6234, 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.
\(\displaystyle \int \frac {x^3 \left (c^2 x^2+1\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {\int -\frac {\cosh ^6\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\cosh ^6\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^4}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {\int \left (\frac {\sinh \left (\frac {9 a}{b}-\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 (a+b \text {arcsinh}(c x))}+\frac {3 \sinh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 (a+b \text {arcsinh}(c x))}-\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}-\frac {3 \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3}{128} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{32} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{256} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{256} \sinh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{128} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {1}{32} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {3}{256} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{256} \cosh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{b c^4}\) |
((3*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/128 + (CoshIntegral[(3 *(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/32 - (3*CoshIntegral[(7*(a + b*Ar cSinh[c*x]))/b]*Sinh[(7*a)/b])/256 - (CoshIntegral[(9*(a + b*ArcSinh[c*x]) )/b]*Sinh[(9*a)/b])/256 - (3*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b ])/128 - (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/32 + (3* Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/256 + (Cosh[(9*a)/ b]*SinhIntegral[(9*(a + b*ArcSinh[c*x]))/b])/256)/(b*c^4)
3.4.72.3.1 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 = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {{\mathrm e}^{\frac {9 a}{b}} \operatorname {Ei}_{1}\left (9 \,\operatorname {arcsinh}\left (c x \right )+\frac {9 a}{b}\right )+3 \,{\mathrm e}^{\frac {7 a}{b}} \operatorname {Ei}_{1}\left (7 \,\operatorname {arcsinh}\left (c x \right )+\frac {7 a}{b}\right )-8 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )-6 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )+6 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )+8 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )-3 \,{\mathrm e}^{-\frac {7 a}{b}} \operatorname {Ei}_{1}\left (-7 \,\operatorname {arcsinh}\left (c x \right )-\frac {7 a}{b}\right )-{\mathrm e}^{-\frac {9 a}{b}} \operatorname {Ei}_{1}\left (-9 \,\operatorname {arcsinh}\left (c x \right )-\frac {9 a}{b}\right )}{512 c^{4} b}\) | \(197\) |
1/512*(exp(9*a/b)*Ei(1,9*arcsinh(c*x)+9*a/b)+3*exp(7*a/b)*Ei(1,7*arcsinh(c *x)+7*a/b)-8*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)-6*exp(a/b)*Ei(1,arcsinh (c*x)+a/b)+6*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)+8*exp(-3*a/b)*Ei(1,-3*arcsi nh(c*x)-3*a/b)-3*exp(-7*a/b)*Ei(1,-7*arcsinh(c*x)-7*a/b)-exp(-9*a/b)*Ei(1, -9*arcsinh(c*x)-9*a/b))/c^4/b
\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^{3} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Exception generated. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^3\,{\left (c^2\,x^2+1\right )}^{5/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]