\(\int \frac {x^2 (1+c^2 x^2)^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [405]

Optimal result

Integrand size = 27, antiderivative size = 219 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c^3}-\frac {3 \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b^2 c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.40 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {16 b c^2 x^2+32 b c^4 x^4+16 b c^6 x^6-(a+b \text {arcsinh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+4 (a+b \text {arcsinh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \text {arcsinh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+b \text {arcsinh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 b \text {arcsinh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 b \text {arcsinh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b^2 c^3 (a+b \text {arcsinh}(c x))} \] Input:

Integrate[(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-1/16*(16*b*c^2*x^2 + 32*b*c^4*x^4 + 16*b*c^6*x^6 - (a + b*ArcSinh[c*x])*C 
oshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] + 4*(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])]*Sinh[(6*a)/b] + a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + Arc 
Sinh[c*x])] + b*ArcSinh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c 
*x])] - 4*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 4*b*ArcSi 
nh[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])])/(b^2*c^3*(a + b*ArcSinh[c*x]))
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6229, 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.

\(\displaystyle \int \frac {x^2 \left (c^2 x^2+1\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6229

\(\displaystyle \frac {2 \int \frac {x \left (c^2 x^2+1\right )}{a+b \text {arcsinh}(c x)}dx}{b c}+\frac {6 c \int \frac {x^3 \left (c^2 x^2+1\right )}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {6 \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^3}+\frac {2 \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {6 \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {6 \int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}-\frac {3 \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 (a+b \text {arcsinh}(c x))}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{4 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^3}+\frac {6 \left (\frac {3}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

Input:

Int[(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-((x^2*(1 + c^2*x^2)^2)/(b*c*(a + b*ArcSinh[c*x]))) + (2*(-1/4*(CoshIntegr 
al[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b]) - (CoshIntegral[(4*(a + b*Ar 
cSinh[c*x]))/b]*Sinh[(4*a)/b])/8 + (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*A 
rcSinh[c*x]))/b])/4 + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x])) 
/b])/8))/(b^2*c^3) + (6*((3*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[ 
(2*a)/b])/32 - (CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/32 
 - (3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/32 + (Cosh[( 
6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/32))/(b^2*c^3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

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 6229

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] + (-Simp[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] - Simp[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 6234

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]
 

Maple [A] (verified)

Time = 4.96 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {32 b \,c^{6} x^{6}+64 b \,c^{4} x^{4}+32 b \,c^{2} x^{2}+4 \,{\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )-3 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {expIntegral}_{1}\left (6 \,\operatorname {arcsinh}\left (x c \right )+\frac {6 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )-4 \,{\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )+{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )-{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )+3 \,{\mathrm e}^{-\frac {6 a}{b}} \operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arcsinh}\left (x c \right )-\frac {6 a}{b}\right ) b \,\operatorname {arcsinh}\left (x c \right )+4 \,{\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right ) a -3 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {expIntegral}_{1}\left (6 \,\operatorname {arcsinh}\left (x c \right )+\frac {6 a}{b}\right ) a -4 \,{\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right ) a +{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right ) a -{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right ) a +3 \,{\mathrm e}^{-\frac {6 a}{b}} \operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arcsinh}\left (x c \right )-\frac {6 a}{b}\right ) a}{32 c^{3} b^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}\)	\(369\)

Input:

int(x^2*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/32*(32*b*c^6*x^6+64*b*c^4*x^4+32*b*c^2*x^2+4*exp(-4*a/b)*Ei(1,-4*arcsin 
h(x*c)-4*a/b)*b*arcsinh(x*c)-3*exp(6*a/b)*Ei(1,6*arcsinh(x*c)+6*a/b)*b*arc 
sinh(x*c)-4*exp(4*a/b)*Ei(1,4*arcsinh(x*c)+4*a/b)*b*arcsinh(x*c)+exp(2*a/b 
)*Ei(1,2*arcsinh(x*c)+2*a/b)*b*arcsinh(x*c)-exp(-2*a/b)*Ei(1,-2*arcsinh(x* 
c)-2*a/b)*b*arcsinh(x*c)+3*exp(-6*a/b)*Ei(1,-6*arcsinh(x*c)-6*a/b)*b*arcsi 
nh(x*c)+4*exp(-4*a/b)*Ei(1,-4*arcsinh(x*c)-4*a/b)*a-3*exp(6*a/b)*Ei(1,6*ar 
csinh(x*c)+6*a/b)*a-4*exp(4*a/b)*Ei(1,4*arcsinh(x*c)+4*a/b)*a+exp(2*a/b)*E 
i(1,2*arcsinh(x*c)+2*a/b)*a-exp(-2*a/b)*Ei(1,-2*arcsinh(x*c)-2*a/b)*a+3*ex 
p(-6*a/b)*Ei(1,-6*arcsinh(x*c)-6*a/b)*a)/c^3/b^2/(a+b*arcsinh(x*c))
 

Fricas [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^2*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas" 
)
 

Output:

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

Sympy [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \] Input:

integrate(x**2*(c**2*x**2+1)**(3/2)/(a+b*asinh(c*x))**2,x)
 

Output:

Integral(x**2*(c**2*x**2 + 1)**(3/2)/(a + b*asinh(c*x))**2, x)
 

Maxima [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^2*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima" 
)
 

Output:

-((c^4*x^6 + 2*c^2*x^4 + x^2)*(c^2*x^2 + 1) + (c^5*x^7 + 2*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(((6*c^5*x^6 + 7*c^3*x^4 + c*x^2)*(c^2*x^2 + 1)^(3/2) + 
 2*(6*c^6*x^7 + 11*c^4*x^5 + 6*c^2*x^3 + x)*(c^2*x^2 + 1) + 3*(2*c^7*x^8 + 
 5*c^5*x^6 + 4*c^3*x^4 + 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]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^2*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")
 

Output:

integrate((c^2*x^2 + 1)^(3/2)*x^2/(b*arcsinh(c*x) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{3/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((x^2*(c^2*x^2 + 1)^(3/2))/(a + b*asinh(c*x))^2,x)
 

Output:

int((x^2*(c^2*x^2 + 1)^(3/2))/(a + b*asinh(c*x))^2, x)
 

Reduce [F]

\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{4}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{2}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \] Input:

int(x^2*(c^2*x^2+1)^(3/2)/(a+b*asinh(c*x))^2,x)
 

Output:

int((sqrt(c**2*x**2 + 1)*x**4)/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a* 
*2),x)*c**2 + int((sqrt(c**2*x**2 + 1)*x**2)/(asinh(c*x)**2*b**2 + 2*asinh 
(c*x)*a*b + a**2),x)