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

Optimal. Leaf size=78 \[ \text{Unintegrable}\left (\frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )},x\right )+\frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b} \]

[Out]

(CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/b - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/b + Unintegrab
le[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]

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Rubi [A]  time = 0.398859, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.86417, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.267, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\arcsin \left ( cx \right ) \right ) }\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)),x)

[Out]

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

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Maxima [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 )} 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)),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Integral(sqrt(-(c*x - 1)*(c*x + 1))/(x*(a + b*asin(c*x))), 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 )} 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)),x, algorithm="giac")

[Out]

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