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

Optimal. Leaf size=163 \[ -\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSin[a*x]]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Fresnel
S[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(a*Sqrt[1 - a^2*x^2]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[
ArcSin[a*x]])/Sqrt[Pi]])/(a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.133859, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4659, 4723, 4406, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSin[a*x]]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Fresnel
S[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(a*Sqrt[1 - a^2*x^2]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[
ArcSin[a*x]])/Sqrt[Pi]])/(a*Sqrt[1 - a^2*x^2])

Rule 4659

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*
(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^Frac
Part[p])/(b*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x],
x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [C]  time = 0.376242, size = 211, normalized size = 1.29 \[ -\frac{c \sqrt{c-a^2 c x^2} e^{-4 i \sin ^{-1}(a x)} \left (-2 e^{4 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )-2 e^{4 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )+16 \sqrt{\pi } e^{4 i \sin ^{-1}(a x)} \sqrt{\sin ^{-1}(a x)} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )+6 e^{4 i \sin ^{-1}(a x)}+e^{8 i \sin ^{-1}(a x)}+8 e^{4 i \sin ^{-1}(a x)} \cos \left (2 \sin ^{-1}(a x)\right )+1\right )}{8 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*Sqrt[c - a^2*c*x^2]*(1 + 6*E^((4*I)*ArcSin[a*x]) + E^((8*I)*ArcSin[a*x]) + 8*E^((4*I)*ArcSin[a*x])*Cos[2*A
rcSin[a*x]] + 16*E^((4*I)*ArcSin[a*x])*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] - 2
*E^((4*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 2*E^((4*I)*ArcSin[a*x])*Sqrt[I*
ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]]))/(8*a*E^((4*I)*ArcSin[a*x])*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])

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Maple [F]  time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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