3.329 \(\int \frac{(1-c^2 x^2)^{3/2}}{x (a+b \sin ^{-1}(c x))} \, dx\)

Optimal. Leaf size=139 \[ \text{Unintegrable}\left (\frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )},x\right )+\frac{5 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{4 b}+\frac{\sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b}-\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{4 b}-\frac{\cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b} \]

[Out]

(5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b) + (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/
(4*b) - (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b) - (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x
]))/b])/(4*b) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]

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Rubi [A]  time = 0.752415, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])),x]

[Out]

(5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(4*b) + (CosIntegral[(3*a)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b])/(4*b)
- (5*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(4*b) - (Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b
) + Defer[Int][1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]

Rubi steps

\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{3/2}}{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{2 c^2 x}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}+\frac{c^4 x^3}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}\right ) \, dx\\ &=-\left (\left (2 c^2\right ) \int \frac{x}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\right )+c^4 \int \frac{x^3}{\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\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\right )+\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 ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\left (\left (2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\right )+\left (2 \sin \left (\frac{a}{b}\right )\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+\operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 (a+b x)}-\frac{\sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{2 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{b}-\frac{2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{\sin (x)}{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{2 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{b}-\frac{2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b}+\frac{1}{4} \left (3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{4} \cos \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{4} \left (3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac{1}{4} \sin \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 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{5 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{4 b}+\frac{\text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac{3 a}{b}\right )}{4 b}-\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{4 b}-\frac{\cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 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 = 2.91594, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])), x]

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Maple [A]  time = 0.268, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\arcsin \left ( cx \right ) \right ) } \left ( -{c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x)

[Out]

int((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\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)^(3/2)/x/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((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{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{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)^(3/2)/x/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(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{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{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)**(3/2)/x/(a+b*asin(c*x)),x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(x*(a + b*asin(c*x))), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\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)^(3/2)/x/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((b*arcsin(c*x) + a)*x), x)