\(\int \frac {(d+e x^2)^2}{a+b \text {arcsinh}(c x)} \, dx\) [156]

Optimal result

Integrand size = 20, antiderivative size = 388 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}+\frac {d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c^3}-\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}+\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 b c^3}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}-\frac {d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c^3}+\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.65 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {2 \left (8 c^4 d^2-4 c^2 d e+e^2\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+\left (8 c^2 d-3 e\right ) e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-16 c^4 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+8 c^2 d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-2 e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-8 c^2 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{16 b c^5} \] Input:

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

Output:

(2*(8*c^4*d^2 - 4*c^2*d*e + e^2)*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x] 
] + (8*c^2*d - 3*e)*e*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + 
 e^2*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 16*c^4*d^2*Sinh[ 
a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 8*c^2*d*e*Sinh[a/b]*SinhIntegral[a 
/b + ArcSinh[c*x]] - 2*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 8* 
c^2*d*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 3*e^2*Sinh[(3 
*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - e^2*Sinh[(5*a)/b]*SinhIntegr 
al[5*(a/b + ArcSinh[c*x])])/(16*b*c^5)
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6208, 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.

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

\(\Big \downarrow \) 6208

\(\displaystyle \int \left (\frac {d^2}{a+b \text {arcsinh}(c x)}+\frac {2 d e x^2}{a+b \text {arcsinh}(c x)}+\frac {e^2 x^4}{a+b \text {arcsinh}(c x)}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (d*e*Cosh[a/b 
]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(2*b*c^3) + (e^2*Cosh[a/b]*CoshInt 
egral[(a + b*ArcSinh[c*x])/b])/(8*b*c^5) + (d*e*Cosh[(3*a)/b]*CoshIntegral 
[(3*(a + b*ArcSinh[c*x]))/b])/(2*b*c^3) - (3*e^2*Cosh[(3*a)/b]*CoshIntegra 
l[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (e^2*Cosh[(5*a)/b]*CoshIntegra 
l[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (d^2*Sinh[a/b]*SinhIntegral[(a 
 + b*ArcSinh[c*x])/b])/(b*c) + (d*e*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[ 
c*x])/b])/(2*b*c^3) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b]) 
/(8*b*c^5) - (d*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/ 
(2*b*c^3) + (3*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b]) 
/(16*b*c^5) - (e^2*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b]) 
/(16*b*c^5)
 

Defintions of rubi rules used

rule 2009

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

rule 6208

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 
 0 || IGtQ[n, 0])
 

Maple [A] (verified)

Time = 3.52 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {-\frac {e^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )}{32 c^{4} b}-\frac {e^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )}{32 c^{4} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) d e}{4 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) e^{2}}{16 c^{4} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) d e}{4 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) e^{2}}{16 c^{4} b}-\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right ) d}{4 c^{2} b}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{32 c^{4} b}-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right ) d}{4 c^{2} b}+\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{32 c^{4} b}}{c}\)	\(380\)
default	\(\frac {-\frac {e^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )}{32 c^{4} b}-\frac {e^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )}{32 c^{4} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) d e}{4 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) e^{2}}{16 c^{4} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) d e}{4 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) e^{2}}{16 c^{4} b}-\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right ) d}{4 c^{2} b}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{32 c^{4} b}-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right ) d}{4 c^{2} b}+\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{32 c^{4} b}}{c}\)	\(380\)

Input:

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

Output:

1/c*(-1/32/c^4*e^2/b*exp(5*a/b)*Ei(1,5*arcsinh(x*c)+5*a/b)-1/32/c^4*e^2/b* 
exp(-5*a/b)*Ei(1,-5*arcsinh(x*c)-5*a/b)-1/2/b*exp(a/b)*Ei(1,arcsinh(x*c)+a 
/b)*d^2+1/4/c^2/b*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)*d*e-1/16/c^4/b*exp(a/b)* 
Ei(1,arcsinh(x*c)+a/b)*e^2-1/2/b*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)*d^2+1/4 
/c^2/b*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)*d*e-1/16/c^4/b*exp(-a/b)*Ei(1,-ar 
csinh(x*c)-a/b)*e^2-1/4/c^2*e/b*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)*d+3/ 
32/c^4*e^2/b*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)-1/4/c^2*e/b*exp(-3*a/b) 
*Ei(1,-3*arcsinh(x*c)-3*a/b)*d+3/32/c^4*e^2/b*exp(-3*a/b)*Ei(1,-3*arcsinh( 
x*c)-3*a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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