\(\int \frac {(1+c^2 x^2)^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [430]

Optimal result

Integrand size = 24, antiderivative size = 216 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {15 \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c}-\frac {3 \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c}-\frac {3 \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c}+\frac {15 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}+\frac {3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c} \]

[Out]

-(c^2*x^2+1)^3/b/c/(a+b*arcsinh(c*x))+15/16*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c*x))/b)/b^2/c+3/4*cosh(4*a/b)*Shi(
4*(a+b*arcsinh(c*x))/b)/b^2/c+3/16*cosh(6*a/b)*Shi(6*(a+b*arcsinh(c*x))/b)/b^2/c-15/16*Chi(2*(a+b*arcsinh(c*x)
)/b)*sinh(2*a/b)/b^2/c-3/4*Chi(4*(a+b*arcsinh(c*x))/b)*sinh(4*a/b)/b^2/c-3/16*Chi(6*(a+b*arcsinh(c*x))/b)*sinh
(6*a/b)/b^2/c

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5790, 5819, 5556, 3384, 3379, 3382} \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {15 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}-\frac {3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}+\frac {15 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}+\frac {3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}-\frac {\left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))} \]

[In]

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

[Out]

-((1 + c^2*x^2)^3/(b*c*(a + b*ArcSinh[c*x]))) - (15*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(1
6*b^2*c) - (3*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c) - (3*CoshIntegral[(6*(a + b*Ar
cSinh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c) + (15*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(16*b^
2*c) + (3*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c) + (3*Cosh[(6*a)/b]*SinhIntegral[(6
*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c)

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 5790

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

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 {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {(6 c) \int \frac {x \left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b} \\ & = -\frac {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {6 \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} \\ & = -\frac {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {6 \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} \\ & = -\frac {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {3 \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}-\frac {3 \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}-\frac {15 \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} \\ & = -\frac {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\left (15 \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}+\frac {\left (3 \cosh \left (\frac {4 a}{b}\right )\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}+\frac {\left (3 \cosh \left (\frac {6 a}{b}\right )\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}-\frac {\left (15 \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}-\frac {\left (3 \sinh \left (\frac {4 a}{b}\right )\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}-\frac {\left (3 \sinh \left (\frac {6 a}{b}\right )\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} \\ & = -\frac {\left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {15 \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c}-\frac {3 \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c}-\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}+\frac {15 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c}+\frac {3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c}+\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.44 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {16 b+48 b c^2 x^2+48 b c^4 x^4+16 b c^6 x^6+15 (a+b \text {arcsinh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+12 (a+b \text {arcsinh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \text {arcsinh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )-15 a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-15 b \text {arcsinh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-12 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-12 b \text {arcsinh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 b \text {arcsinh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b^2 c (a+b \text {arcsinh}(c x))} \]

[In]

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

[Out]

-1/16*(16*b + 48*b*c^2*x^2 + 48*b*c^4*x^4 + 16*b*c^6*x^6 + 15*(a + b*ArcSinh[c*x])*CoshIntegral[2*(a/b + ArcSi
nh[c*x])]*Sinh[(2*a)/b] + 12*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 3*a*Cos
hIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(
6*a)/b] - 15*a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 15*b*ArcSinh[c*x]*Cosh[(2*a)/b]*SinhIntegr
al[2*(a/b + ArcSinh[c*x])] - 12*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 12*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*A
rcSinh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(b^2*c*(a + b*ArcSinh[c*x]))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.73

method	result	size
default	\(\frac {-32 b \,c^{6} x^{6}-96 b \,c^{4} x^{4}-96 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 ) b \,\operatorname {arcsinh}\left (c x \right )+12 \,{\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 )+15 \,{\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 )-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 )-15 \,{\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 )-12 \,{\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 )+3 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {Ei}_{1}\left (6 \,\operatorname {arcsinh}\left (c x \right )+\frac {6 a}{b}\right ) a +12 \,{\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (c x \right )+\frac {4 a}{b}\right ) a +15 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 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 -15 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right ) a -12 \,{\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}{32 c \,b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\)	\(374\)

[In]

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

[Out]

1/32*(-32*b*c^6*x^6-96*b*c^4*x^4-96*b*c^2*x^2+3*exp(6*a/b)*Ei(1,6*arcsinh(c*x)+6*a/b)*b*arcsinh(c*x)+12*exp(4*
a/b)*Ei(1,4*arcsinh(c*x)+4*a/b)*b*arcsinh(c*x)+15*exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)*b*arcsinh(c*x)-3*exp(-
6*a/b)*Ei(1,-6*arcsinh(c*x)-6*a/b)*b*arcsinh(c*x)-15*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)*b*arcsinh(c*x)-12
*exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)*b*arcsinh(c*x)+3*exp(6*a/b)*Ei(1,6*arcsinh(c*x)+6*a/b)*a+12*exp(4*a/b
)*Ei(1,4*arcsinh(c*x)+4*a/b)*a+15*exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)*a-3*exp(-6*a/b)*Ei(1,-6*arcsinh(c*x)-6
*a/b)*a-15*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)*a-12*exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)*a-32*b)/c/b^2/
(a+b*arcsinh(c*x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-((c^6*x^6 + 3*c^4*x^4 + 3*c^2*x^2 + 1)*(c^2*x^2 + 1) + (c^7*x^7 + 3*c^5*x^5 + 3*c^3*x^3 + c*x)*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(((6*c^6*x^6 + 11*c^4*x^4 + 4*c^2*x^2 - 1)*(c^2*x^2 + 1)^(3/2) + 6*(2
*c^7*x^7 + 5*c^5*x^5 + 4*c^3*x^3 + c*x)*(c^2*x^2 + 1) + (6*c^8*x^8 + 19*c^6*x^6 + 21*c^4*x^4 + 9*c^2*x^2 + 1)*
sqrt(c^2*x^2 + 1))/(a*b*c^4*x^4 + (c^2*x^2 + 1)*a*b*c^2*x^2 + 2*a*b*c^2*x^2 + a*b + (b^2*c^4*x^4 + (c^2*x^2 +
1)*b^2*c^2*x^2 + 2*b^2*c^2*x^2 + b^2 + 2*(b^2*c^3*x^3 + b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1
)) + 2*(a*b*c^3*x^3 + a*b*c*x)*sqrt(c^2*x^2 + 1)), x)

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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