3.7.42 \(\int \frac {1}{(d+e x^2)^2 \sqrt {a+b \sinh ^{-1}(c x)}} \, dx\) [642]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \sinh ^{-1}(c x)}},x\right ) \]

[Out]

Unintegrable(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^(1/2),x)

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \sinh ^{-1}(c x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b \sinh ^{-1}(c x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right )^{2} \sqrt {a +b \arcsinh \left (c x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^(1/2),x)

[Out]

int(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^(1/2),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(a+b*arcsinh(c*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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}\,{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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