3.472 \(\int \frac{1}{\sqrt{1+b x^2} \sin ^{-1}(\sqrt{1+b x^2})} \, dx\)

Optimal. Leaf size=30 \[ \frac{\sqrt{-b x^2} \log \left (\sin ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]

[Out]

(Sqrt[-(b*x^2)]*Log[ArcSin[Sqrt[1 + b*x^2]]])/(b*x)

________________________________________________________________________________________

Rubi [A]  time = 0.0605774, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {4834, 4639} \[ \frac{\sqrt{-b x^2} \log \left (\sin ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + b*x^2]*ArcSin[Sqrt[1 + b*x^2]]),x]

[Out]

(Sqrt[-(b*x^2)]*Log[ArcSin[Sqrt[1 + b*x^2]]])/(b*x)

Rule 4834

Int[ArcSin[Sqrt[1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[1 + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[-(b*x^2)]/(b*x), Subst
[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]

Rule 4639

Int[1/(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[Log[a + b*ArcSin[c*x]]
/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1+b x^2} \sin ^{-1}\left (\sqrt{1+b x^2}\right )} \, dx &=\frac{\sqrt{-b x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)} \, dx,x,\sqrt{1+b x^2}\right )}{b x}\\ &=\frac{\sqrt{-b x^2} \log \left (\sin ^{-1}\left (\sqrt{1+b x^2}\right )\right )}{b x}\\ \end{align*}

Mathematica [A]  time = 0.0232356, size = 26, normalized size = 0.87 \[ -\frac{x \log \left (\sin ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{\sqrt{-b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + b*x^2]*ArcSin[Sqrt[1 + b*x^2]]),x]

[Out]

-((x*Log[ArcSin[Sqrt[1 + b*x^2]]])/Sqrt[-(b*x^2)])

________________________________________________________________________________________

Maple [F]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arcsin \left ( \sqrt{b{x}^{2}+1} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arcsin((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x)

[Out]

int(1/arcsin((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} + 1} \arcsin \left (\sqrt{b x^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^2 + 1)*arcsin(sqrt(b*x^2 + 1))), x)

________________________________________________________________________________________

Fricas [A]  time = 1.91662, size = 68, normalized size = 2.27 \begin{align*} \frac{\sqrt{-b x^{2}} \log \left (-\arcsin \left (\sqrt{b x^{2} + 1}\right )\right )}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

sqrt(-b*x^2)*log(-arcsin(sqrt(b*x^2 + 1)))/(b*x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asin((b*x**2+1)**(1/2))/(b*x**2+1)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x**2 + 1)*asin(sqrt(b*x**2 + 1))), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} + 1} \arcsin \left (\sqrt{b x^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin((b*x^2+1)^(1/2))/(b*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 + 1)*arcsin(sqrt(b*x^2 + 1))), x)