\(\int \frac {(d+c^2 d x^2)^2}{a+b \text {arcsinh}(c x)} \, dx\) [28]

Optimal result

Integrand size = 23, antiderivative size = 201 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {5 d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c}+\frac {5 d^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c}+\frac {d^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c}-\frac {5 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c}-\frac {5 d^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c}-\frac {d^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c} \] Output:

5/8*d^2*cosh(a/b)*Chi((a+b*arcsinh(c*x))/b)/b/c+5/16*d^2*cosh(3*a/b)*Chi(3 
*(a+b*arcsinh(c*x))/b)/b/c+1/16*d^2*cosh(5*a/b)*Chi(5*(a+b*arcsinh(c*x))/b 
)/b/c-5/8*d^2*sinh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b/c-5/16*d^2*sinh(3*a/b) 
*Shi(3*(a+b*arcsinh(c*x))/b)/b/c-1/16*d^2*sinh(5*a/b)*Shi(5*(a+b*arcsinh(c 
*x))/b)/b/c
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {d^2 \left (10 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-10 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-5 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )\right )}{16 b c} \] Input:

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

Output:

(d^2*(10*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + 5*Cosh[(3*a)/b]*Cosh 
Integral[3*(a/b + ArcSinh[c*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + Arc 
Sinh[c*x])] - 10*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 5*Sinh[(3*a) 
/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a 
/b + ArcSinh[c*x])]))/(16*b*c)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6206, 3042, 3793, 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.

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

\(\Big \downarrow \) 6206

\(\displaystyle \frac {d^2 \int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^5}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {d^2 \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \left (\frac {5}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {5}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {5}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b c}\)

Input:

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

Output:

(d^2*((5*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/8 + (5*Cosh[(3*a) 
/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/16 + (Cosh[(5*a)/b]*CoshInte 
gral[(5*(a + b*ArcSinh[c*x]))/b])/16 - (5*Sinh[a/b]*SinhIntegral[(a + b*Ar 
cSinh[c*x])/b])/8 - (5*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x])) 
/b])/16 - (Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/16))/(b 
*c)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6206

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Subst[Int 
[x^n*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, 0]
 

Maple [A] (verified)

Time = 2.89 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {-\frac {d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )}{32 b}-\frac {5 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{32 b}-\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{16 b}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{16 b}-\frac {5 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{32 b}-\frac {d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )}{32 b}}{c}\)	\(182\)
default	\(\frac {-\frac {d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )}{32 b}-\frac {5 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{32 b}-\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{16 b}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{16 b}-\frac {5 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{32 b}-\frac {d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )}{32 b}}{c}\)	\(182\)

Input:

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

Output:

1/c*(-1/32*d^2/b*exp(5*a/b)*Ei(1,5*arcsinh(x*c)+5*a/b)-5/32*d^2/b*exp(3*a/ 
b)*Ei(1,3*arcsinh(x*c)+3*a/b)-5/16*d^2/b*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)-5 
/16*d^2/b*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)-5/32*d^2/b*exp(-3*a/b)*Ei(1,-3 
*arcsinh(x*c)-3*a/b)-1/32*d^2/b*exp(-5*a/b)*Ei(1,-5*arcsinh(x*c)-5*a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

d**2*(Integral(2*c**2*x**2/(a + b*asinh(c*x)), x) + Integral(c**4*x**4/(a 
+ b*asinh(c*x)), x) + Integral(1/(a + b*asinh(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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