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

Optimal. Leaf size=104 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac{\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2}-\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2}-\frac{1-c^2 x^2}{b c x \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((1 - c^2*x^2)/(b*c*x*(a + b*ArcSin[c*x]))) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/b^2 - (Sin[a/b]*S
inIntegral[(a + b*ArcSin[c*x])/b])/b^2 - Unintegrable[1/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)

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Rubi [A]  time = 0.203671, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((1 - c^2*x^2)/(b*c*x*(a + b*ArcSin[c*x]))) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/b^2 - (Sin[a/b]*S
inIntegral[(a + b*ArcSin[c*x])/b])/b^2 - Defer[Int][1/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)

Rubi steps

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

Mathematica [A]  time = 10.226, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.532, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*x^2 - (b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x)*integrate((c^2*x^2 + 1)/(b^2*c*x^2*a
rctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2), x) - 1)/(b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1)) + a*b*c*x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1}}{b^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b x \arcsin \left (c x\right ) + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*x^2 + 1)/(b^2*x*arcsin(c*x)^2 + 2*a*b*x*arcsin(c*x) + a^2*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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