\(\int \frac {d+e x^2}{a+b \text {arcsinh}(c x)} \, dx\) [157]

Optimal result

Integrand size = 18, antiderivative size = 180 \[ \int \frac {d+e x^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}+\frac {e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}+\frac {e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}-\frac {e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int \frac {d+e x^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {\left (4 c^2 d-e\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 c^2 d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{4 b c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 6208

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (e*Cosh[a/b]*Co 
shIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) + (e*Cosh[(3*a)/b]*CoshInteg 
ral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3) - (d*Sinh[a/b]*SinhIntegral[(a 
+ b*ArcSinh[c*x])/b])/(b*c) + (e*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x 
])/b])/(4*b*c^3) - (e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/ 
b])/(4*b*c^3)
 

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 = 2.95 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) d}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) e}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) d}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) e}{8 c^{2} b}}{c}\)	\(178\)
default	\(\frac {-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) d}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right ) e}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) d}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right ) e}{8 c^{2} b}}{c}\)	\(178\)

Input:

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

Output:

1/c*(-1/8*e/c^2/b*exp(-3*a/b)*Ei(1,-3*arcsinh(x*c)-3*a/b)-1/8*e/c^2/b*exp( 
3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)-1/2/b*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)*d+ 
1/8/c^2/b*exp(a/b)*Ei(1,arcsinh(x*c)+a/b)*e-1/2/b*exp(-a/b)*Ei(1,-arcsinh( 
x*c)-a/b)*d+1/8/c^2/b*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)*e)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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