\(\int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\) [337]

Optimal result

Integrand size = 21, antiderivative size = 40 \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^4}{4 a \sqrt {c+a^2 c x^2}} \]

[Out]

1/4*arcsinh(a*x)^4*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5783} \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^4}{4 a \sqrt {a^2 c x^2+c}} \]

[In]

Int[ArcSinh[a*x]^3/Sqrt[c + a^2*c*x^2],x]

[Out]

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

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^4}{4 a \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^4}{4 a \sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

Integrate[ArcSinh[a*x]^3/Sqrt[c + a^2*c*x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98

method	result	size
default	\(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{4}}{4 \sqrt {a^{2} x^{2}+1}\, a c}\)	\(39\)

[In]

int(arcsinh(a*x)^3/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c*arcsinh(a*x)^4

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^3/sqrt(a^2*c*x^2 + c), x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(asinh(a*x)**3/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(asinh(a*x)**3/sqrt(c*(a**2*x**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.35 \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\operatorname {arsinh}\left (a x\right )^{4}}{4 \, a \sqrt {c}} \]

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

1/4*arcsinh(a*x)^4/(a*sqrt(c))

Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^3/sqrt(a^2*c*x^2 + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

int(asinh(a*x)^3/(c + a^2*c*x^2)^(1/2),x)

[Out]

int(asinh(a*x)^3/(c + a^2*c*x^2)^(1/2), x)