3.5.66 \(\int \frac {(c-a^2 c x^2)^{3/2}}{\sqrt {\text {ArcSin}(a x)}} \, dx\) [466]

Optimal. Leaf size=170 \[ \frac {3 c \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}}{4 a \sqrt {1-a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{8 a \sqrt {1-a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{2 a \sqrt {1-a^2 x^2}} \]

[Out]

1/16*c*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/
2)+1/2*c*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+3/4*c*(-a^2
*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)/a/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4753, 3393, 3385, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{8 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{2 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3385

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 3393

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 3433

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

Rule 4753

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 182, normalized size = 1.07 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)} \left (24 \sqrt {\text {ArcSin}(a x)^2}-4 \sqrt {2} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcSin}(a x)\right )-4 \sqrt {2} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcSin}(a x)\right )-\sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcSin}(a x)\right )-\sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcSin}(a x)\right )\right )}{32 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(24*Sqrt[ArcSin[a*x]^2] - 4*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (-
2*I)*ArcSin[a*x]] - 4*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma
[1/2, (-4*I)*ArcSin[a*x]] - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]]))/(32*a*Sqrt[1 - a^2*x^2]*Sqr
t[ArcSin[a*x]^2])

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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\sqrt {\arcsin \left (a x \right )}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a^2*c*x^2)^(3/2)/asin(a*x)^(1/2),x)

[Out]

int((c - a^2*c*x^2)^(3/2)/asin(a*x)^(1/2), x)

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