3.477 \(\int \frac{(c-a^2 c x^2)^{3/2}}{\sin ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=206 \[ -\frac{4 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(3*a*ArcSin[a*x]^(3/2)) + (16*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^
2])/(3*Sqrt[ArcSin[a*x]]) - (4*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a
*Sqrt[1 - a^2*x^2]) - (8*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*a*Sqrt[1
- a^2*x^2])

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Rubi [A]  time = 0.29656, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4659, 4721, 4661, 3312, 3304, 3352, 4723, 4406} \[ -\frac{4 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(5/2),x]

[Out]

(-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(3*a*ArcSin[a*x]^(3/2)) + (16*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^
2])/(3*Sqrt[ArcSin[a*x]]) - (4*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a
*Sqrt[1 - a^2*x^2]) - (8*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*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 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(f*m*d^IntPar
t[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/
2)*(a + b*ArcSin[c*x])^(n + 1), x], x] + Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n +
1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x])
/; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]

Rule 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*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] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(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 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[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])

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]

Rubi steps

\begin{align*} \int \frac{\left (c-a^2 c x^2\right )^{3/2}}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{\left (8 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (1-a^2 x^2\right )}{\sin ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{\sin ^{-1}(a x)}} \, dx}{3 \sqrt{1-a^2 x^2}}+\frac{\left (64 a^2 c \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2 \sqrt{1-a^2 x^2}}{\sqrt{\sin ^{-1}(a x)}} \, dx}{3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a \sqrt{1-a^2 x^2}}+\frac{\left (64 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 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}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 a \sqrt{1-a^2 x^2}}+\frac{\left (64 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{8 \sqrt{x}}-\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 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}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (8 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{\left (8 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 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}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 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}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{4 c \sqrt{2 \pi } \sqrt{c-a^2 c x^2} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a \sqrt{1-a^2 x^2}}-\frac{8 c \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 1.33268, size = 251, normalized size = 1.22 \[ \frac{c \sqrt{c-a^2 c x^2} \left (-16 \sqrt{2} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )-16 \sqrt{2} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )-16 \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )-16 \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )+16 a^2 x^2+64 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-e^{-4 i \sin ^{-1}(a x)}-e^{4 i \sin ^{-1}(a x)}+8 i e^{-4 i \sin ^{-1}(a x)} \sin ^{-1}(a x)-8 i e^{4 i \sin ^{-1}(a x)} \sin ^{-1}(a x)-14\right )}{24 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(5/2),x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-14 - E^((-4*I)*ArcSin[a*x]) - E^((4*I)*ArcSin[a*x]) + 16*a^2*x^2 + ((8*I)*ArcSin[a*x]
)/E^((4*I)*ArcSin[a*x]) - (8*I)*E^((4*I)*ArcSin[a*x])*ArcSin[a*x] + 64*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 16*
Sqrt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcSin[a*x]] - 16*Sqrt[2]*(I*ArcSin[a*x])^(3/2)*Gamma[1/2,
(2*I)*ArcSin[a*x]] - 16*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 16*(I*ArcSin[a*x])^(3/2)*Gam
ma[1/2, (4*I)*ArcSin[a*x]]))/(24*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))

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Maple [F]  time = 0.183, 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{5}{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)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(5/2),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)^(5/2),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)^(5/2),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)**(5/2),x)

[Out]

Timed out

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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{5}{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)^(5/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)/arcsin(a*x)^(5/2), x)