3.5.77 \(\int \frac {(c-a^2 c x^2)^{3/2}}{\text {ArcSin}(a x)^{5/2}} \, dx\) [477]

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

[Out]

-2/3*(-a^2*c*x^2+c)^(3/2)*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)-8/3*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)-4/3*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)+16/3*c*x*(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(1
/2)

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Rubi [A]
time = 0.20, antiderivative size = 206, normalized size of antiderivative = 1.00, 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 = {4751, 4799, 4753, 3393, 3385, 3433, 4809, 4491} \begin {gather*} -\frac {4 \sqrt {2 \pi } c \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(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 {\text {ArcSin}(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 \text {ArcSin}(a x)^{3/2}}+\frac {16 c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {ArcSin}(a x)}} \end {gather*}

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 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 4491

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 4751

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)/(b*(n + 1)))*Simp[(d + e*x^2)
^p/(1 - c^2*x^2)^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 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]

Rule 4799

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/(b*c*(n
+ 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 4809

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.89, size = 251, normalized size = 1.22 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-14-e^{-4 i \text {ArcSin}(a x)}-e^{4 i \text {ArcSin}(a x)}+16 a^2 x^2+8 i e^{-4 i \text {ArcSin}(a x)} \text {ArcSin}(a x)-8 i e^{4 i \text {ArcSin}(a x)} \text {ArcSin}(a x)+64 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)-16 \sqrt {2} (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcSin}(a x)\right )-16 \sqrt {2} (i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcSin}(a x)\right )-16 (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcSin}(a x)\right )-16 (i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcSin}(a x)\right )\right )}{24 a \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arcsin \left (a x \right )^{\frac {5}{2}}}\, dx\]

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 = 0.00, 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)^(5/2),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)^(5/2),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}}}{\operatorname {asin}^{\frac {5}{2}}{\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)**(5/2),x)

[Out]

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)/asin(a*x)**(5/2), 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)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,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)^(5/2),x)

[Out]

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

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