3.5.71 \(\int \frac {(c-a^2 c x^2)^{5/2}}{\text {ArcSin}(a x)^{3/2}} \, dx\) [471]

Optimal. Leaf size=237 \[ -\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {ArcSin}(a x)}}-\frac {3 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{2 a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {3 \pi } \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {15 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}} \]

[Out]

-3/4*c^2*FresnelS(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)-15/8*c^2*FresnelS(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)-1/8*c^
2*FresnelS(2*3^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-2*
(-a^2*c*x^2+c)^(5/2)*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4751, 4809, 4491, 3386, 3432} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{2 a \sqrt {1-a^2 x^2}}-\frac {\sqrt {3 \pi } c^2 \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {15 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {ArcSin}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(5/2))/(a*Sqrt[ArcSin[a*x]]) - (3*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Fre
snelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a*Sqrt[1 - a^2*x^2]) - (c^2*Sqrt[3*Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[
2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(8*a*Sqrt[1 - a^2*x^2]) - (15*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sq
rt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])

Rule 3386

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[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 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 )^{5/2}}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (12 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (1-a^2 x^2\right )^2}{\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 )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (12 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos ^5(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 )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (12 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {5 \sin (2 x)}{32 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}+\frac {\sin (6 x)}{32 \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 )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (6 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a \sqrt {1-a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (6 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {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 (15 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{4 a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {3 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {3 \pi } \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {15 c^2 \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.73, size = 404, normalized size = 1.70 \begin {gather*} -\frac {c^2 e^{-6 i \text {ArcSin}(a x)} \sqrt {c-a^2 c x^2} \left (1+6 e^{2 i \text {ArcSin}(a x)}+15 e^{4 i \text {ArcSin}(a x)}+20 e^{6 i \text {ArcSin}(a x)}+15 e^{8 i \text {ArcSin}(a x)}+6 e^{10 i \text {ArcSin}(a x)}+e^{12 i \text {ArcSin}(a x)}+64 e^{6 i \text {ArcSin}(a x)} \sqrt {\pi } \sqrt {\text {ArcSin}(a x)} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )+\sqrt {2} e^{6 i \text {ArcSin}(a x)} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcSin}(a x)\right )+\sqrt {2} e^{6 i \text {ArcSin}(a x)} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcSin}(a x)\right )-12 e^{6 i \text {ArcSin}(a x)} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcSin}(a x)\right )-12 e^{6 i \text {ArcSin}(a x)} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcSin}(a x)\right )-\sqrt {6} e^{6 i \text {ArcSin}(a x)} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},-6 i \text {ArcSin}(a x)\right )-\sqrt {6} e^{6 i \text {ArcSin}(a x)} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {1}{2},6 i \text {ArcSin}(a x)\right )\right )}{32 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(c^2*Sqrt[c - a^2*c*x^2]*(1 + 6*E^((2*I)*ArcSin[a*x]) + 15*E^((4*I)*ArcSin[a*x]) + 20*E^((6*I)*ArcSin[a*
x]) + 15*E^((8*I)*ArcSin[a*x]) + 6*E^((10*I)*ArcSin[a*x]) + E^((12*I)*ArcSin[a*x]) + 64*E^((6*I)*ArcSin[a*x])*
Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] + Sqrt[2]*E^((6*I)*ArcSin[a*x])*Sqrt[(-I)*
ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] + Sqrt[2]*E^((6*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (2*
I)*ArcSin[a*x]] - 12*E^((6*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 12*E^((6*I)
*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]] - Sqrt[6]*E^((6*I)*ArcSin[a*x])*Sqrt[(-I)*ArcS
in[a*x]]*Gamma[1/2, (-6*I)*ArcSin[a*x]] - Sqrt[6]*E^((6*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (6*I)*A
rcSin[a*x]]))/(a*E^((6*I)*ArcSin[a*x])*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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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)^(5/2)/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(5/2)/arcsin(a*x)^(3/2), 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 )}^{5/2}}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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