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

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

[Out]

2*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a/(a^2*c*x^2+c)^(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} \[ \frac {2 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{a \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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