\(\int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx\) [380]

Optimal result

Integrand size = 27, antiderivative size = 144 \[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c^5}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b c^5}+\frac {3 \log (a+b \text {arcsinh}(c x))}{8 b c^5}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c^5}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b c^5} \] Output:

-1/2*cosh(2*a/b)*Chi(2*(a+b*arcsinh(c*x))/b)/b/c^5+1/8*cosh(4*a/b)*Chi(4*( 
a+b*arcsinh(c*x))/b)/b/c^5+3/8*ln(a+b*arcsinh(c*x))/b/c^5+1/2*sinh(2*a/b)* 
Shi(2*(a+b*arcsinh(c*x))/b)/b/c^5-1/8*sinh(4*a/b)*Shi(4*(a+b*arcsinh(c*x)) 
/b)/b/c^5
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=-\frac {4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 \log (a+b \text {arcsinh}(c x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{8 b c^5} \] Input:

Integrate[x^4/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]
 

Output:

-1/8*(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] - Cosh[(4*a)/b] 
*CoshIntegral[4*(a/b + ArcSinh[c*x])] - 3*Log[a + b*ArcSinh[c*x]] - 4*Sinh 
[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + Sinh[(4*a)/b]*SinhIntegra 
l[4*(a/b + ArcSinh[c*x])])/(b*c^5)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6234, 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.

\(\displaystyle \int \frac {x^4}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))} \, dx\)

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^4}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^5}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 (a+b \text {arcsinh}(c x))}+\frac {3}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {3}{8} \log (a+b \text {arcsinh}(c x))}{b c^5}\)

Input:

Int[x^4/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]
 

Output:

(-1/2*(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]) + (Cosh[(4* 
a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8 + (3*Log[a + b*ArcSinh[c 
*x]])/8 + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/2 - (Si 
nh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8)/(b*c^5)
 

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 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 = 3.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {{\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right )-4 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )-4 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )+{\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right )-6 \ln \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c^{5} b}\)	\(115\)

Input:

int(x^4/(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)-4*exp(2*a/b)*Ei(1,2*arcsinh(x 
*c)+2*a/b)-4*exp(-2*a/b)*Ei(1,-2*arcsinh(x*c)-2*a/b)+exp(-4*a/b)*Ei(1,-4*a 
rcsinh(x*c)-4*a/b)-6*ln(a+b*arcsinh(x*c)))/c^5/b
 

Fricas [F]

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

integrate(x^4/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

integral(sqrt(c^2*x^2 + 1)*x^4/(a*c^2*x^2 + (b*c^2*x^2 + b)*arcsinh(c*x) + 
 a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^4/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

integrate(x^4/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)), x)
 

Giac [F]

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

integrate(x^4/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x)),x, algorithm="giac")
 

Output:

integrate(x^4/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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