\(\int \frac {x^3 (1+c^2 x^2)^{3/2}}{a+b \text {arcsinh}(c x)} \, dx\) [357]

Optimal result

Integrand size = 27, antiderivative size = 245 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {3 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b c^4}+\frac {3 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b c^4}-\frac {\text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b c^4}-\frac {\text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4} \] Output:

3/64*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b/c^4+3/64*Chi(3*(a+b*arcsinh(c*x 
))/b)*sinh(3*a/b)/b/c^4-1/64*Chi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b/c^4 
-1/64*Chi(7*(a+b*arcsinh(c*x))/b)*sinh(7*a/b)/b/c^4-3/64*cosh(a/b)*Shi((a+ 
b*arcsinh(c*x))/b)/b/c^4-3/64*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b/c^ 
4+1/64*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c*x))/b)/b/c^4+1/64*cosh(7*a/b)*Shi( 
7*(a+b*arcsinh(c*x))/b)/b/c^4
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {3 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-\text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b c^4} \] Input:

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

Output:

(3*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*CoshIntegral[3*(a/b + Ar 
cSinh[c*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5* 
a)/b] - CoshIntegral[7*(a/b + ArcSinh[c*x])]*Sinh[(7*a)/b] - 3*Cosh[a/b]*S 
inhIntegral[a/b + ArcSinh[c*x]] - 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + Ar 
cSinh[c*x])] + Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + Cosh[( 
7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b*c^4)
 

Rubi [A] (verified)

Time = 1.15 (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.

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

\(\Big \downarrow \) 6234

\(\displaystyle \frac {\int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^4}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {\int \left (\frac {\sinh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {3 \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {3}{64} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {3}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{64} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {3}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{b c^4}\)

Input:

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

Output:

((3*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/64 + (3*CoshIntegral[( 
3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/64 - (CoshIntegral[(5*(a + b*Arc 
Sinh[c*x]))/b]*Sinh[(5*a)/b])/64 - (CoshIntegral[(7*(a + b*ArcSinh[c*x]))/ 
b]*Sinh[(7*a)/b])/64 - (3*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/ 
64 - (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/64 + (Cosh 
[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/64 + (Cosh[(7*a)/b]*Si 
nhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/64)/(b*c^4)
 

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 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 = 1.91 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {{\mathrm e}^{\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (7 \,\operatorname {arcsinh}\left (x c \right )+\frac {7 a}{b}\right )+{\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )-3 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )-3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )+3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )+3 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )-{\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )-{\mathrm e}^{-\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (-7 \,\operatorname {arcsinh}\left (x c \right )-\frac {7 a}{b}\right )}{128 c^{4} b}\)	\(196\)

Input:

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

Output:

1/128*(exp(7*a/b)*Ei(1,7*arcsinh(x*c)+7*a/b)+exp(5*a/b)*Ei(1,5*arcsinh(x*c 
)+5*a/b)-3*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)-3*exp(a/b)*Ei(1,arcsinh(x 
*c)+a/b)+3*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)+3*exp(-3*a/b)*Ei(1,-3*arcsinh 
(x*c)-3*a/b)-exp(-5*a/b)*Ei(1,-5*arcsinh(x*c)-5*a/b)-exp(-7*a/b)*Ei(1,-7*a 
rcsinh(x*c)-7*a/b))/c^4/b
 

Fricas [F]

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

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

Output:

integral((c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)/(b*arcsinh(c*x) + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{5}}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) c^{2}+\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{3}}{\mathit {asinh} \left (c x \right ) b +a}d x \] Input:

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

Output:

int((sqrt(c**2*x**2 + 1)*x**5)/(asinh(c*x)*b + a),x)*c**2 + int((sqrt(c**2 
*x**2 + 1)*x**3)/(asinh(c*x)*b + a),x)