\(\int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx\) [54]

Optimal result

Integrand size = 10, antiderivative size = 82 \[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=-\frac {x^6 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}-\frac {5 \text {Shi}(\text {arcsinh}(a x))}{64 a^7}+\frac {27 \text {Shi}(3 \text {arcsinh}(a x))}{64 a^7}-\frac {25 \text {Shi}(5 \text {arcsinh}(a x))}{64 a^7}+\frac {7 \text {Shi}(7 \text {arcsinh}(a x))}{64 a^7} \] Output:

-x^6*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)-5/64*Shi(arcsinh(a*x))/a^7+27/64*Shi 
(3*arcsinh(a*x))/a^7-25/64*Shi(5*arcsinh(a*x))/a^7+7/64*Shi(7*arcsinh(a*x) 
)/a^7
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=-\frac {64 a^6 x^6 \sqrt {1+a^2 x^2}+5 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))-27 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))+25 \text {arcsinh}(a x) \text {Shi}(5 \text {arcsinh}(a x))-7 \text {arcsinh}(a x) \text {Shi}(7 \text {arcsinh}(a x))}{64 a^7 \text {arcsinh}(a x)} \] Input:

Integrate[x^6/ArcSinh[a*x]^2,x]
 

Output:

-1/64*(64*a^6*x^6*Sqrt[1 + a^2*x^2] + 5*ArcSinh[a*x]*SinhIntegral[ArcSinh[ 
a*x]] - 27*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]] + 25*ArcSinh[a*x]*Sin 
hIntegral[5*ArcSinh[a*x]] - 7*ArcSinh[a*x]*SinhIntegral[7*ArcSinh[a*x]])/( 
a^7*ArcSinh[a*x])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6193, 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.

\(\displaystyle \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx\)

\(\Big \downarrow \) 6193

\(\displaystyle \frac {\int \left (-\frac {5 a x}{64 \text {arcsinh}(a x)}+\frac {27 \sinh (3 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}-\frac {25 \sinh (5 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}+\frac {7 \sinh (7 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^7}-\frac {x^6 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {5}{64} \text {Shi}(\text {arcsinh}(a x))+\frac {27}{64} \text {Shi}(3 \text {arcsinh}(a x))-\frac {25}{64} \text {Shi}(5 \text {arcsinh}(a x))+\frac {7}{64} \text {Shi}(7 \text {arcsinh}(a x))}{a^7}-\frac {x^6 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\)

Input:

Int[x^6/ArcSinh[a*x]^2,x]
 

Output:

-((x^6*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + ((-5*SinhIntegral[ArcSinh[a* 
x]])/64 + (27*SinhIntegral[3*ArcSinh[a*x]])/64 - (25*SinhIntegral[5*ArcSin 
h[a*x]])/64 + (7*SinhIntegral[7*ArcSinh[a*x]])/64)/a^7
 

Defintions of rubi rules used

rule 2009

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

rule 6193

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si 
mp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- 
a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi 
nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 
1]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {\frac {5 \sqrt {a^{2} x^{2}+1}}{64 \,\operatorname {arcsinh}\left (x a \right )}-\frac {5 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (x a \right )\right )}{64}-\frac {9 \cosh \left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{64 \,\operatorname {arcsinh}\left (x a \right )}+\frac {27 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{64}+\frac {5 \cosh \left (5 \,\operatorname {arcsinh}\left (x a \right )\right )}{64 \,\operatorname {arcsinh}\left (x a \right )}-\frac {25 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (x a \right )\right )}{64}-\frac {\cosh \left (7 \,\operatorname {arcsinh}\left (x a \right )\right )}{64 \,\operatorname {arcsinh}\left (x a \right )}+\frac {7 \,\operatorname {Shi}\left (7 \,\operatorname {arcsinh}\left (x a \right )\right )}{64}}{a^{7}}\)	\(104\)
default	\(\frac {\frac {5 \sqrt {a^{2} x^{2}+1}}{64 \,\operatorname {arcsinh}\left (x a \right )}-\frac {5 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (x a \right )\right )}{64}-\frac {9 \cosh \left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{64 \,\operatorname {arcsinh}\left (x a \right )}+\frac {27 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{64}+\frac {5 \cosh \left (5 \,\operatorname {arcsinh}\left (x a \right )\right )}{64 \,\operatorname {arcsinh}\left (x a \right )}-\frac {25 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (x a \right )\right )}{64}-\frac {\cosh \left (7 \,\operatorname {arcsinh}\left (x a \right )\right )}{64 \,\operatorname {arcsinh}\left (x a \right )}+\frac {7 \,\operatorname {Shi}\left (7 \,\operatorname {arcsinh}\left (x a \right )\right )}{64}}{a^{7}}\)	\(104\)

Input:

int(x^6/arcsinh(x*a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^7*(5/64/arcsinh(x*a)*(a^2*x^2+1)^(1/2)-5/64*Shi(arcsinh(x*a))-9/64/arc 
sinh(x*a)*cosh(3*arcsinh(x*a))+27/64*Shi(3*arcsinh(x*a))+5/64/arcsinh(x*a) 
*cosh(5*arcsinh(x*a))-25/64*Shi(5*arcsinh(x*a))-1/64/arcsinh(x*a)*cosh(7*a 
rcsinh(x*a))+7/64*Shi(7*arcsinh(x*a)))
 

Fricas [F]

\[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^6/arcsinh(a*x)^2,x, algorithm="fricas")
 

Output:

integral(x^6/arcsinh(a*x)^2, x)
 

Sympy [F]

\[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{6}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \] Input:

integrate(x**6/asinh(a*x)**2,x)
 

Output:

Integral(x**6/asinh(a*x)**2, x)
 

Maxima [F]

\[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^6/arcsinh(a*x)^2,x, algorithm="maxima")
 

Output:

-(a^3*x^9 + a*x^7 + (a^2*x^8 + x^6)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^ 
2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2 + 1))) + integrate((7*a^5*x^1 
0 + 14*a^3*x^8 + 7*a*x^6 + (7*a^3*x^8 + 5*a*x^6)*(a^2*x^2 + 1) + (14*a^4*x 
^9 + 19*a^2*x^7 + 6*x^5)*sqrt(a^2*x^2 + 1))/((a^5*x^4 + (a^2*x^2 + 1)*a^3* 
x^2 + 2*a^3*x^2 + 2*(a^4*x^3 + a^2*x)*sqrt(a^2*x^2 + 1) + a)*log(a*x + sqr 
t(a^2*x^2 + 1))), x)
 

Giac [F]

\[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^6/arcsinh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(x^6/arcsinh(a*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^6}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \] Input:

int(x^6/asinh(a*x)^2,x)
 

Output:

int(x^6/asinh(a*x)^2, x)
 

Reduce [F]

\[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{6}}{\mathit {asinh} \left (a x \right )^{2}}d x \] Input:

int(x^6/asinh(a*x)^2,x)
 

Output:

int(x**6/asinh(a*x)**2,x)