\(\int \frac {x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)} \, dx\) [351]

Optimal result

Integrand size = 27, antiderivative size = 82 \[ \int \frac {x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b c^3}-\frac {\log (a+b \text {arcsinh}(c x))}{8 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b c^3} \] Output:

1/8*cosh(4*a/b)*Chi(4*(a+b*arcsinh(c*x))/b)/b/c^3-1/8*ln(a+b*arcsinh(c*x)) 
/b/c^3-1/8*sinh(4*a/b)*Shi(4*(a+b*arcsinh(c*x))/b)/b/c^3
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\log (a+b \text {arcsinh}(c x))-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{8 b c^3} \] Input:

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

Output:

(Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] - Log[a + b*ArcSinh[c* 
x]] - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(8*b*c^3)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6234, 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.

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8 - Log[a + b*Ar 
cSinh[c*x]]/8 - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8 
)/(b*c^3)
 

Defintions of rubi rules used

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 = 2.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {{\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right )+{\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right )+2 \ln \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c^{3} b}\)	\(67\)

Input:

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

Output:

-1/16*(exp(4*a/b)*Ei(1,4*arcsinh(x*c)+4*a/b)+exp(-4*a/b)*Ei(1,-4*arcsinh(x 
*c)-4*a/b)+2*ln(a+b*arcsinh(x*c)))/c^3/b
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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