3.7.61 \(\int \frac {1}{\sqrt {d+e x^2} (a+b \sinh ^{-1}(c x))^2} \, dx\) [661]

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

[Out]

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

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Rubi [A]
time = 0.03, 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}{\sqrt {d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 4.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}

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

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 = 0.00, 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/(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(x^2*e + d)*a*b*c^2*x + (sqrt(c^2*x^2 + 1)*sqrt(
x^2*e + d)*b^2*c^2*x + (b^2*c^3*x^2 + b^2*c)*sqrt(x^2*e + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*
b*c)*sqrt(x^2*e + 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*x^4*e + a*b*c^3*d*x^2)*(c^2*x^2 + 1)*sq
rt(x^2*e + d) + 2*(a*b*c^4*x^5*e + a*b*c^2*d*x + (a*b*c^4*d + a*b*c^2*e)*x^3)*sqrt(c^2*x^2 + 1)*sqrt(x^2*e + d
) + ((b^2*c^3*x^4*e + b^2*c^3*d*x^2)*(c^2*x^2 + 1)*sqrt(x^2*e + d) + 2*(b^2*c^4*x^5*e + b^2*c^2*d*x + (b^2*c^4
*d + b^2*c^2*e)*x^3)*sqrt(c^2*x^2 + 1)*sqrt(x^2*e + d) + (b^2*c^5*x^6*e + (b^2*c^5*d + 2*b^2*c^3*e)*x^4 + b^2*
c*d + (2*b^2*c^3*d + b^2*c*e)*x^2)*sqrt(x^2*e + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^6*e + (a*b*c^5*d
+ 2*a*b*c^3*e)*x^4 + a*b*c*d + (2*a*b*c^3*d + a*b*c*e)*x^2)*sqrt(x^2*e + d)), x)

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

[Out]

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

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

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 = 0.00, 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/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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