\(\int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx\) [51]

Optimal result

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

[Out]

-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-x^6*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5778, 3379} \[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=-\frac {5 \text {Shi}(\text {arcsinh}(a x))}{64 a^7}+\frac {27 \text {Shi}(3 \text {arcsinh}(a x))}{64 a^7}-\frac {25 \text {Shi}(5 \text {arcsinh}(a x))}{64 a^7}+\frac {7 \text {Shi}(7 \text {arcsinh}(a x))}{64 a^7}-\frac {x^6 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)} \]

[In]

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

[Out]

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

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 5778

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^6 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \left (-\frac {5 \sinh (x)}{64 x}+\frac {27 \sinh (3 x)}{64 x}-\frac {25 \sinh (5 x)}{64 x}+\frac {7 \sinh (7 x)}{64 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^7} \\ & = -\frac {x^6 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}-\frac {5 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7}+\frac {7 \text {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7}-\frac {25 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7}+\frac {27 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\text {arcsinh}(a x)^2} \, dx=-\frac {64 a^6 x^6 \sqrt {1+a^2 x^2}+5 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))-27 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))+25 \text {arcsinh}(a x) \text {Shi}(5 \text {arcsinh}(a x))-7 \text {arcsinh}(a x) \text {Shi}(7 \text {arcsinh}(a x))}{64 a^7 \text {arcsinh}(a x)} \]

[In]

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

[Out]

-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]*SinhIntegral[5*ArcSinh[a*x]] - 7*ArcSinh[a*x]*SinhIntegral[7*ArcSinh[a*x]])
/(a^7*ArcSinh[a*x])

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^6/arcsinh(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**6/asinh(a*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

-(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^10 + 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 + sqrt(a^2*x^2 + 1))), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^6/arcsinh(a*x)^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^6/asinh(a*x)^2, x)