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

Optimal. Leaf size=176 \[ -\frac{\text{Unintegrable}\left (\frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac{9 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{4 b^2}-\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2}-\frac{9 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{4 b^2}-\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2}-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((1 - c^2*x^2)^2/(b*c*x*(a + b*ArcSin[c*x]))) - (9*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2) - (3*
Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2) - (9*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])
/(4*b^2) - (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2) - Unintegrable[(1 - c^2*x^2)/(x^2*(
a + b*ArcSin[c*x])), x]/(b*c)

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Rubi [A]  time = 0.403718, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((1 - c^2*x^2)^2/(b*c*x*(a + b*ArcSin[c*x]))) - (9*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(4*b^2) - (3*Cos[
(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b^2) - (9*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(4*b^2) -
(3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b^2) - Defer[Int][(1 - c^2*x^2)/(x^2*(a + b*ArcSin[c
*x])), x]/(b*c)

Rubi steps

\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac{\int \frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac{(3 c) \int \frac{1-c^2 x^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\int \frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac{3 \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 (a+b x)}+\frac{\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\int \frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac{\int \frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac{\int \frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac{\left (9 \cos \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 )}{4 b}-\frac{\left (3 \cos \left (\frac{3 a}{b}\right )\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 )}{4 b}-\frac{\left (9 \sin \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 )}{4 b}-\frac{\left (3 \sin \left (\frac{3 a}{b}\right )\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 )}{4 b}\\ &=-\frac{\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac{9 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{4 b^2}-\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2}-\frac{9 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{4 b^2}-\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2}-\frac{\int \frac{1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ \end{align*}

Mathematica [A]  time = 10.3348, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.343, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}} \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))^2,x)

[Out]

int((-c^2*x^2+1)^(3/2)/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^{4} x^{4} - 2 \, 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 (3 \, c^{4} \int \frac{x^{4}}{b x^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a x^{2}}\,{d x} - 2 \, 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}{b x^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a 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)^(3/2)/x/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-(c^4*x^4 - 2*c^2*x^2 - (b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x)*integrate((3*c^4*x^4 -
2*c^2*x^2 - 1)/(b^2*c*x^2*arctan2(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{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{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)^(3/2)/x/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(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{\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 )^{2}}\, 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))**2,x)

[Out]

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

[Out]

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