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

Optimal. Leaf size=227 \[ -\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]

[Out]

(3*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/8 + (x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]])/4 + (c*Sqrt[c -
a^2*c*x^2]*ArcSin[a*x]^(3/2))/(4*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi
]*Sqrt[ArcSin[a*x]]])/(64*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]]
)/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.282997, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4649, 4647, 4641, 4635, 4406, 12, 3305, 3351, 4723} \[ -\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/8 + (x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]])/4 + (c*Sqrt[c -
a^2*c*x^2]*ArcSin[a*x]^(3/2))/(4*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi
]*Sqrt[ArcSin[a*x]]])/(64*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]]
)/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])

Rule 4649

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

Rule 4647

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

Rule 4641

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

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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])

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} (3 c) \int \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \, dx-\frac{\left (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}{8 \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{8 \sqrt{1-a^2 x^2}}-\frac{\left (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 )}{8 a \sqrt{1-a^2 x^2}}-\frac{\left (3 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{16 \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (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 )}{8 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\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 )}{64 a \sqrt{1-a^2 x^2}}-\frac{\left (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 )}{32 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (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 )}{32 a \sqrt{1-a^2 x^2}}-\frac{\left (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 )}{16 a \sqrt{1-a^2 x^2}}-\frac{\left (3 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 )}{32 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\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 )}{64 a \sqrt{1-a^2 x^2}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}-\frac{\left (3 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 )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\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 )}{64 a \sqrt{1-a^2 x^2}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.21993, size = 166, normalized size = 0.73 \[ \frac{c \sqrt{c-a^2 c x^2} \left (8 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}(a x)\right )+8 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},4 i \sin ^{-1}(a x)\right )+32 \sin ^{-1}(a x)^2\right )}{128 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*(32*ArcSin[a*x]^2 + 8*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + 8
*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-4*I)*ArcSin[a
*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (4*I)*ArcSin[a*x]]))/(128*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])

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Maple [F]  time = 0.197, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{\arcsin \left ( ax \right ) }\, 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)^(1/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2),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)^(1/2),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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(1/2),x)

[Out]

Timed out

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

[Out]

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