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

Optimal. Leaf size=288 \[ -\frac{2 d^2 \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}},x\right )}{b c}-\frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{3 \sqrt{\pi } d^2 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{b^{3/2}}-\frac{3 \sqrt{\pi } d^2 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}} \]

[Out]

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

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Rubi [A]  time = 1.4632, 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 = {} \[ \int \frac{\left (d-c^2 d x^2\right )^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (2 d^2\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (8 c d^2\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (8 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \left (-\frac{2 c^2}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}+\frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}+\frac{c^4 x^2}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (8 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{a+b x}}+\frac{\cos (2 x)}{2 \sqrt{a+b x}}+\frac{\cos (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}+\frac{\left (4 c d^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}-\frac{\left (2 c^3 d^2\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}+\frac{2 d^2 \sqrt{a+b \sin ^{-1}(c x)}}{b^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}+\frac{2 d^2 \sqrt{a+b \sin ^{-1}(c x)}}{b^2}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}-\frac{\cos (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (4 d^2 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (d^2 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (4 d^2 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (d^2 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (8 d^2 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}-\frac{\left (2 d^2 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}-\frac{\left (8 d^2 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}-\frac{\left (2 d^2 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{d^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{4 d^2 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2}}-\frac{4 d^2 \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{b^{3/2}}-\frac{d^2 \sqrt{\frac{\pi }{2}} S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{b^{3/2}}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}+\frac{\left (d^2 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}+\frac{\left (d^2 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{d^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{4 d^2 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2}}-\frac{4 d^2 \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{b^{3/2}}-\frac{d^2 \sqrt{\frac{\pi }{2}} S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{b^{3/2}}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}+\frac{\left (2 d^2 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}+\frac{\left (2 d^2 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{d^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{3 d^2 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2}}-\frac{3 d^2 \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{b^{3/2}}-\frac{d^2 \sqrt{\frac{\pi }{2}} S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{b^{3/2}}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ \end{align*}

Mathematica [A]  time = 3.5083, size = 0, normalized size = 0. \[ \int \frac{\left (d-c^2 d x^2\right )^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.404, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{c}^{2}d{x}^{2}+d \right ) ^{2}}{x} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int - \frac{2 c^{2} x^{2}}{a x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int \frac{c^{4} x^{4}}{a x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int \frac{1}{a x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**2/x/(a+b*asin(c*x))**(3/2),x)

[Out]

d**2*(Integral(-2*c**2*x**2/(a*x*sqrt(a + b*asin(c*x)) + b*x*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c
**4*x**4/(a*x*sqrt(a + b*asin(c*x)) + b*x*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(1/(a*x*sqrt(a + b*as
in(c*x)) + b*x*sqrt(a + b*asin(c*x))*asin(c*x)), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")

[Out]

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