3.504 \(\int \frac {1}{\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5677, 5675} \[ -\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2)),x]

[Out]

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

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rule 5677

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/S
qrt[d + e*x^2], Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e,
c^2*d] &&  !GtQ[d, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 1.00 \[ -\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2)),x]

[Out]

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

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fricas [A]  time = 0.43, size = 57, normalized size = 1.42 \[ -\frac {2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} + a c\right )} \sqrt {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)/((a^3*c*x^2 + a*c)*sqrt(log(a*x + sqrt(a^2*x^2 + 1))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^(3/2)), x)

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maple [A]  time = 0.06, size = 36, normalized size = 0.90 \[ -\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {\arcsinh \left (a x \right )}\, a \sqrt {c \left (a^{2} x^{2}+1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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