3.7.57 \(\int \frac {1}{\sqrt {d+e x^2} (a+b \sinh ^{-1}(c x))} \, dx\) [657]

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

[Out]

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

________________________________________________________________________________________

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 )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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 ) \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))/(e*x**2+d)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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 )\,\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))*(d + e*x^2)^(1/2)),x)

[Out]

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

________________________________________________________________________________________