\(\int \frac {x^4 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)} \, dx\) [349]

Optimal result

Integrand size = 27, antiderivative size = 206 \[ \int \frac {x^4 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)} \, dx=-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^5}-\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^5}+\frac {\log (a+b \text {arcsinh}(c x))}{16 b c^5}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^5}+\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^5} \] Output:

-1/32*cosh(2*a/b)*Chi(2*(a+b*arcsinh(c*x))/b)/b/c^5-1/16*cosh(4*a/b)*Chi(4 
*(a+b*arcsinh(c*x))/b)/b/c^5+1/32*cosh(6*a/b)*Chi(6*(a+b*arcsinh(c*x))/b)/ 
b/c^5+1/16*ln(a+b*arcsinh(c*x))/b/c^5+1/32*sinh(2*a/b)*Shi(2*(a+b*arcsinh( 
c*x))/b)/b/c^5+1/16*sinh(4*a/b)*Shi(4*(a+b*arcsinh(c*x))/b)/b/c^5-1/32*sin 
h(6*a/b)*Shi(6*(a+b*arcsinh(c*x))/b)/b/c^5
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.74 \[ \int \frac {x^4 \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-2 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+2 \log (a+b \text {arcsinh}(c x))+\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+2 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{32 b c^5} \] Input:

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

Output:

(-(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])]) - 2*Cosh[(4*a)/b]*C 
oshIntegral[4*(a/b + ArcSinh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + 
ArcSinh[c*x])] + 2*Log[a + b*ArcSinh[c*x]] + Sinh[(2*a)/b]*SinhIntegral[2* 
(a/b + ArcSinh[c*x])] + 2*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x] 
)] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(32*b*c^5)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.83, 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.

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\int \left (\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 (a+b \text {arcsinh}(c x))}+\frac {1}{16 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \log (a+b \text {arcsinh}(c x))}{b c^5}\)

Input:

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

Output:

(-1/32*(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]) - (Cosh[(4 
*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/16 + (Cosh[(6*a)/b]*CoshI 
ntegral[(6*(a + b*ArcSinh[c*x]))/b])/32 + Log[a + b*ArcSinh[c*x]]/16 + (Si 
nh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/32 + (Sinh[(4*a)/b]* 
SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/16 - (Sinh[(6*a)/b]*SinhIntegral 
[(6*(a + b*ArcSinh[c*x]))/b])/32)/(b*c^5)
 

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 = 4.90 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {{\mathrm e}^{\frac {6 a}{b}} \operatorname {expIntegral}_{1}\left (6 \,\operatorname {arcsinh}\left (x c \right )+\frac {6 a}{b}\right )-2 \,{\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right )-{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )-{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )-2 \,{\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right )+{\mathrm e}^{-\frac {6 a}{b}} \operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arcsinh}\left (x c \right )-\frac {6 a}{b}\right )-4 \ln \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{64 c^{5} b}\)	\(163\)

Input:

int(x^4*(c^2*x^2+1)^(1/2)/(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

-1/64*(exp(6*a/b)*Ei(1,6*arcsinh(x*c)+6*a/b)-2*exp(4*a/b)*Ei(1,4*arcsinh(x 
*c)+4*a/b)-exp(2*a/b)*Ei(1,2*arcsinh(x*c)+2*a/b)-exp(-2*a/b)*Ei(1,-2*arcsi 
nh(x*c)-2*a/b)-2*exp(-4*a/b)*Ei(1,-4*arcsinh(x*c)-4*a/b)+exp(-6*a/b)*Ei(1, 
-6*arcsinh(x*c)-6*a/b)-4*ln(a+b*arcsinh(x*c)))/c^5/b
 

Fricas [F]

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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