Integrand size = 25, antiderivative size = 183 \[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b c^2}-\frac {3 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b c^2}-\frac {\text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b c^2}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^2}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^2}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^2} \]
1/8*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b/c^2+3/16*cosh(3*a/b)*Shi(3*(a+b* arcsinh(c*x))/b)/b/c^2+1/16*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c*x))/b)/b/c^2- 1/8*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b/c^2-3/16*Chi(3*(a+b*arcsinh(c*x) )/b)*sinh(3*a/b)/b/c^2-1/16*Chi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b/c^2
Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.74 \[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {-2 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b c^2} \]
(-2*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - 3*CoshIntegral[3*(a/b + A rcSinh[c*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5 *a)/b] + 2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3*Cosh[(3*a)/b]*Si nhIntegral[3*(a/b + ArcSinh[c*x])] + Cosh[(5*a)/b]*SinhIntegral[5*(a/b + A rcSinh[c*x])])/(16*b*c^2)
Time = 0.56 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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.
\(\displaystyle \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {\int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {\int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {3}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {3}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{b c^2}\) |
(-1/8*(CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b]) - (3*CoshIntegral[( 3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/16 - (CoshIntegral[(5*(a + b*Arc Sinh[c*x]))/b]*Sinh[(5*a)/b])/16 + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[ c*x])/b])/8 + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/1 6 + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/16)/(b*c^2)
3.4.66.3.1 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 = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {{\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )+3 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )+2 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )-2 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )-3 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )-{\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right )}{32 c^{2} b}\) | \(149\) |
1/32*(exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)+3*exp(3*a/b)*Ei(1,3*arcsinh(c* x)+3*a/b)+2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-2*exp(-a/b)*Ei(1,-arcsinh(c*x) -a/b)-3*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)-exp(-5*a/b)*Ei(1,-5*arcsin h(c*x)-5*a/b))/c^2/b
\[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Exception generated. \[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x\,{\left (c^2\,x^2+1\right )}^{3/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]