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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{\sqrt{d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0473418, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\sqrt{d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 6.22324, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.197, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} x^{3} + c x +{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{\sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} a b c^{2} x +{\left (\sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} b^{2} c^{2} x +{\left (b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt{e x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a b c^{3} x^{2} + a b c\right )} \sqrt{e x^{2} + d}} + \int \frac{c^{5} d x^{4} + 2 \, c^{3} d x^{2} +{\left (c^{2} x^{2} + 1\right )}{\left ({\left (c^{3} d - 2 \, c e\right )} x^{2} - c d\right )} + c d + \sqrt{c^{2} x^{2} + 1}{\left (2 \,{\left (c^{4} d - c^{2} e\right )} x^{3} +{\left (c^{2} d - e\right )} x\right )}}{{\left (a b c^{3} e x^{4} + a b c^{3} d x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} \sqrt{e x^{2} + d} + 2 \,{\left (a b c^{4} e x^{5} + a b c^{2} d x +{\left (c^{4} d + c^{2} e\right )} a b x^{3}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} +{\left ({\left (b^{2} c^{3} e x^{4} + b^{2} c^{3} d x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} \sqrt{e x^{2} + d} + 2 \,{\left (b^{2} c^{4} e x^{5} + b^{2} c^{2} d x +{\left (c^{4} d + c^{2} e\right )} b^{2} x^{3}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} +{\left (b^{2} c^{5} e x^{6} +{\left (c^{5} d + 2 \, c^{3} e\right )} b^{2} x^{4} +{\left (2 \, c^{3} d + c e\right )} b^{2} x^{2} + b^{2} c d\right )} \sqrt{e x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a b c^{5} e x^{6} +{\left (c^{5} d + 2 \, c^{3} e\right )} a b x^{4} +{\left (2 \, c^{3} d + c e\right )} a b x^{2} + a b c d\right )} \sqrt{e x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d)*a*b*c^2*x + (sqrt(c^2*x^2 + 1)*sqrt(
e*x^2 + d)*b^2*c^2*x + (b^2*c^3*x^2 + b^2*c)*sqrt(e*x^2 + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*
b*c)*sqrt(e*x^2 + d)) + integrate((c^5*d*x^4 + 2*c^3*d*x^2 + (c^2*x^2 + 1)*((c^3*d - 2*c*e)*x^2 - c*d) + c*d +
sqrt(c^2*x^2 + 1)*(2*(c^4*d - c^2*e)*x^3 + (c^2*d - e)*x))/((a*b*c^3*e*x^4 + a*b*c^3*d*x^2)*(c^2*x^2 + 1)*sqr
t(e*x^2 + d) + 2*(a*b*c^4*e*x^5 + a*b*c^2*d*x + (c^4*d + c^2*e)*a*b*x^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (
(b^2*c^3*e*x^4 + b^2*c^3*d*x^2)*(c^2*x^2 + 1)*sqrt(e*x^2 + d) + 2*(b^2*c^4*e*x^5 + b^2*c^2*d*x + (c^4*d + c^2*
e)*b^2*x^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (b^2*c^5*e*x^6 + (c^5*d + 2*c^3*e)*b^2*x^4 + (2*c^3*d + c*e)*b
^2*x^2 + b^2*c*d)*sqrt(e*x^2 + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*e*x^6 + (c^5*d + 2*c^3*e)*a*b*x^4 +
(2*c^3*d + c*e)*a*b*x^2 + a*b*c*d)*sqrt(e*x^2 + d)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}}{a^{2} e x^{2} + a^{2} d +{\left (b^{2} e x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b e x^{2} + a b d\right )} \operatorname{arsinh}\left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(a^2*e*x^2 + a^2*d + (b^2*e*x^2 + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*e*x^2 + a*b*d)*arcsi
nh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{e x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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