\(\int \frac {x^2 (1+c^2 x^2)^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [428]

Optimal result

Integrand size = 27, antiderivative size = 281 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3} \]

[Out]

-x^2*(c^2*x^2+1)^3/b/c/(a+b*arcsinh(c*x))-1/16*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c*x))/b)/b^2/c^3+1/8*cosh(4*a/b)
*Shi(4*(a+b*arcsinh(c*x))/b)/b^2/c^3+3/16*cosh(6*a/b)*Shi(6*(a+b*arcsinh(c*x))/b)/b^2/c^3+1/16*cosh(8*a/b)*Shi
(8*(a+b*arcsinh(c*x))/b)/b^2/c^3+1/16*Chi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b^2/c^3-1/8*Chi(4*(a+b*arcsinh(c
*x))/b)*sinh(4*a/b)/b^2/c^3-3/16*Chi(6*(a+b*arcsinh(c*x))/b)*sinh(6*a/b)/b^2/c^3-1/16*Chi(8*(a+b*arcsinh(c*x))
/b)*sinh(8*a/b)/b^2/c^3

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5814, 5819, 5556, 3384, 3379, 3382} \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b^2 c^3}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))} \]

[In]

Int[(x^2*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]

[Out]

-((x^2*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) + (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])
/(16*b^2*c^3) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(8*b^2*c^3) - (3*CoshIntegral[(6*(a +
b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3) - (CoshIntegral[(8*(a + b*ArcSinh[c*x]))/b]*Sinh[(8*a)/b])/(1
6*b^2*c^3) - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3) + (Cosh[(4*a)/b]*SinhIntegr
al[(4*(a + b*ArcSinh[c*x]))/b])/(8*b^2*c^3) + (3*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(16*b
^2*c^3) + (Cosh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

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]

Rule 5814

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[f*(m/(b*c*(
n + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(
n + 1), x], x] - Dist[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(
1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d]
&& LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {2 \int \frac {x \left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b c}+\frac {(8 c) \int \frac {x^3 \left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b} \\ & = -\frac {x^2 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {2 \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3}-\frac {8 \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3} \\ & = -\frac {x^2 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {2 \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{32 x}+\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {5 \sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{32 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3}-\frac {8 \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{128 x}+\frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{64 x}-\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{64 x}-\frac {3 \sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3} \\ & = -\frac {x^2 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3}-\frac {5 \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3} \\ & = -\frac {x^2 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\left (5 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}-\frac {\left (3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}-\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}-\frac {\left (5 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}+\frac {\left (3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3} \\ & = -\frac {x^2 \left (1+c^2 x^2\right )^3}{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 )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.47 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {16 b c^2 x^2+48 b c^4 x^4+48 b c^6 x^6+16 b c^8 x^8-(a+b \text {arcsinh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+2 (a+b \text {arcsinh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \text {arcsinh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \text {Chi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {8 a}{b}\right )+b \text {arcsinh}(c x) \text {Chi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {8 a}{b}\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+b \text {arcsinh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-2 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-2 b \text {arcsinh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 b \text {arcsinh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-a \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-b \text {arcsinh}(c x) \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b^2 c^3 (a+b \text {arcsinh}(c x))} \]

[In]

Integrate[(x^2*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]

[Out]

-1/16*(16*b*c^2*x^2 + 48*b*c^4*x^4 + 48*b*c^6*x^6 + 16*b*c^8*x^8 - (a + b*ArcSinh[c*x])*CoshIntegral[2*(a/b +
ArcSinh[c*x])]*Sinh[(2*a)/b] + 2*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 3*a
*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Si
nh[(6*a)/b] + a*CoshIntegral[8*(a/b + ArcSinh[c*x])]*Sinh[(8*a)/b] + b*ArcSinh[c*x]*CoshIntegral[8*(a/b + ArcS
inh[c*x])]*Sinh[(8*a)/b] + a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + b*ArcSinh[c*x]*Cosh[(2*a)/b]
*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 2*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 2*b*ArcSinh[c
*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])
] - 3*b*ArcSinh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - a*Cosh[(8*a)/b]*SinhIntegral[8*(a/b
+ ArcSinh[c*x])] - b*ArcSinh[c*x]*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcSinh[c*x])])/(b^2*c^3*(a + b*ArcSinh[
c*x]))

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.72

method	result	size
default	\(\frac {-96 b \,c^{6} x^{6}-2 \,{\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 {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 {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 {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 {8 a}{b}} \operatorname {Ei}_{1}\left (-8 \,\operatorname {arcsinh}\left (c x \right )-\frac {8 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )+{\mathrm e}^{\frac {8 a}{b}} \operatorname {Ei}_{1}\left (8 \,\operatorname {arcsinh}\left (c x \right )+\frac {8 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )+3 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {Ei}_{1}\left (6 \,\operatorname {arcsinh}\left (c x \right )+\frac {6 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )-3 \,{\mathrm e}^{-\frac {6 a}{b}} \operatorname {Ei}_{1}\left (-6 \,\operatorname {arcsinh}\left (c x \right )-\frac {6 a}{b}\right ) b \,\operatorname {arcsinh}\left (c x \right )-96 b \,c^{4} x^{4}-32 b \,c^{2} x^{2}+3 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {Ei}_{1}\left (6 \,\operatorname {arcsinh}\left (c x \right )+\frac {6 a}{b}\right ) a -3 \,{\mathrm e}^{-\frac {6 a}{b}} \operatorname {Ei}_{1}\left (-6 \,\operatorname {arcsinh}\left (c x \right )-\frac {6 a}{b}\right ) a +{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right ) a -{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right ) a -{\mathrm e}^{-\frac {8 a}{b}} \operatorname {Ei}_{1}\left (-8 \,\operatorname {arcsinh}\left (c x \right )-\frac {8 a}{b}\right ) a +{\mathrm e}^{\frac {8 a}{b}} \operatorname {Ei}_{1}\left (8 \,\operatorname {arcsinh}\left (c x \right )+\frac {8 a}{b}\right ) a -2 \,{\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 {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (c x \right )+\frac {4 a}{b}\right ) a -32 b \,c^{8} x^{8}}{32 c^{3} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\)	\(484\)

[In]

int(x^2*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/32*(-96*b*c^6*x^6-2*exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)*b*arcsinh(c*x)+exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-
2*a/b)*b*arcsinh(c*x)-exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)*b*arcsinh(c*x)+2*exp(4*a/b)*Ei(1,4*arcsinh(c*x)+4*
a/b)*b*arcsinh(c*x)-exp(-8*a/b)*Ei(1,-8*arcsinh(c*x)-8*a/b)*b*arcsinh(c*x)+exp(8*a/b)*Ei(1,8*arcsinh(c*x)+8*a/
b)*b*arcsinh(c*x)+3*exp(6*a/b)*Ei(1,6*arcsinh(c*x)+6*a/b)*b*arcsinh(c*x)-3*exp(-6*a/b)*Ei(1,-6*arcsinh(c*x)-6*
a/b)*b*arcsinh(c*x)-96*b*c^4*x^4-32*b*c^2*x^2+3*exp(6*a/b)*Ei(1,6*arcsinh(c*x)+6*a/b)*a-3*exp(-6*a/b)*Ei(1,-6*
arcsinh(c*x)-6*a/b)*a+exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)*a-exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)*a-exp(-8
*a/b)*Ei(1,-8*arcsinh(c*x)-8*a/b)*a+exp(8*a/b)*Ei(1,8*arcsinh(c*x)+8*a/b)*a-2*exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)
-4*a/b)*a+2*exp(4*a/b)*Ei(1,4*arcsinh(c*x)+4*a/b)*a-32*b*c^8*x^8)/c^3/b^2/(a+b*arcsinh(c*x))

Fricas [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^6 + 2*c^2*x^4 + x^2)*sqrt(c^2*x^2 + 1)/(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2), x)

Sympy [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate(x**2*(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x))**2,x)

[Out]

Integral(x**2*(c**2*x**2 + 1)**(5/2)/(a + b*asinh(c*x))**2, x)

Maxima [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^8 + 3*c^4*x^6 + 3*c^2*x^4 + x^2)*(c^2*x^2 + 1) + (c^7*x^9 + 3*c^5*x^7 + 3*c^3*x^5 + c*x^3)*sqrt(c^2*x
^2 + 1))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2
*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((8*c^7*x^8 + 17*c^5*x^6 + 10*c^3*x^4 + c*x^2)*(c^2*x^2 + 1)^(3/
2) + 2*(8*c^8*x^9 + 22*c^6*x^7 + 21*c^4*x^5 + 8*c^2*x^3 + x)*(c^2*x^2 + 1) + (8*c^9*x^10 + 27*c^7*x^8 + 33*c^5
*x^6 + 17*c^3*x^4 + 3*c*x^2)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^4 + (c^2*x^2 + 1)*a*b*c^3*x^2 + 2*a*b*c^3*x^2 + a*b
*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2*c^3*x^2 + 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 + b^2*c^2*x)*sqrt(c^2*x
^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^3 + a*b*c^2*x)*sqrt(c^2*x^2 + 1)), x)

Giac [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate((c^2*x^2 + 1)^(5/2)*x^2/(b*arcsinh(c*x) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int((x^2*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x))^2,x)

[Out]

int((x^2*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x))^2, x)