3.5.65 \(\int \frac {(c-a^2 c x^2)^{5/2}}{\sqrt {\text {ArcSin}(a x)}} \, dx\) [465]

Optimal. Leaf size=244 \[ \frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {3 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{3}} \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {15 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}} \]

[Out]

1/96*c^2*(-a^2*c*x^2+c)^(1/2)*FresnelC(2*3^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a/(-a^2*x^2+1)^(
1/2)+3/32*c^2*(-a^2*c*x^2+c)^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a/(-a^2*x^2
+1)^(1/2)+15/32*c^2*(-a^2*c*x^2+c)^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a/(-a^2*x^2+1)^(1/2)+
5/8*c^2*(-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 = 0.11, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {3 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {\frac {\pi }{3}} c^2 \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {5 c^2 \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*c^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(8*a*Sqrt[1 - a^2*x^2]) + (3*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*
FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a*Sqrt[1 - a^2*x^2]) + (c^2*Sqrt[Pi/3]*Sqrt[c - a^2*c*x^2]*Fresn
elC[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(32*a*Sqrt[1 - a^2*x^2]) + (15*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[
(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32*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 )^{5/2}}{\sqrt {\sin ^{-1}(a x)}} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\left (1-a^2 x^2\right )^{5/2}}{\sqrt {\sin ^{-1}(a x)}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos ^6(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {5}{16 \sqrt {x}}+\frac {15 \cos (2 x)}{32 \sqrt {x}}+\frac {3 \cos (4 x)}{16 \sqrt {x}}+\frac {\cos (6 x)}{32 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt {1-a^2 x^2}}\\ &=\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (6 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 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 {\cos (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 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 {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \cos \left (6 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {\left (3 c^2 \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 )}{8 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c^2 \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 )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {3 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{3}} \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {15 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.44, size = 336, normalized size = 1.38 \begin {gather*} \frac {c^2 \sqrt {c-a^2 c x^2} \left (240 \text {ArcSin}(a x) \sqrt {\text {ArcSin}(a x)^2}+3 i \sqrt {2} \left (16 (i \text {ArcSin}(a x))^{3/2}+\sqrt {-i \text {ArcSin}(a x)} \sqrt {\text {ArcSin}(a x)^2}\right ) \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcSin}(a x)\right )-45 i \sqrt {2} (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcSin}(a x)\right )+24 i (i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcSin}(a x)\right )+6 i \sqrt {-i \text {ArcSin}(a x)} \sqrt {\text {ArcSin}(a x)^2} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcSin}(a x)\right )-18 i (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcSin}(a x)\right )-i \sqrt {6} \sqrt {-i \text {ArcSin}(a x)} \sqrt {\text {ArcSin}(a x)^2} \text {Gamma}\left (\frac {1}{2},-6 i \text {ArcSin}(a x)\right )-i \sqrt {6} (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},6 i \text {ArcSin}(a x)\right )\right )}{384 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)} \sqrt {\text {ArcSin}(a x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*Sqrt[c - a^2*c*x^2]*(240*ArcSin[a*x]*Sqrt[ArcSin[a*x]^2] + (3*I)*Sqrt[2]*(16*(I*ArcSin[a*x])^(3/2) + Sqrt
[(-I)*ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2])*Gamma[1/2, (-2*I)*ArcSin[a*x]] - (45*I)*Sqrt[2]*((-I)*ArcSin[a*x])^(3/
2)*Gamma[1/2, (2*I)*ArcSin[a*x]] + (24*I)*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcSin[a*x]] + (6*I)*Sqrt[(-
I)*ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - (18*I)*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2
, (4*I)*ArcSin[a*x]] - I*Sqrt[6]*Sqrt[(-I)*ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2]*Gamma[1/2, (-6*I)*ArcSin[a*x]] - I
*Sqrt[6]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (6*I)*ArcSin[a*x]]))/(384*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]]*S
qrt[ArcSin[a*x]^2])

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\sqrt {\arcsin \left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(5/2)/arcsin(a*x)^(1/2),x)

[Out]

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

[Out]

integrate((-a^2*c*x^2 + c)^(5/2)/sqrt(arcsin(a*x)), 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}}{\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)^(5/2)/asin(a*x)^(1/2),x)

[Out]

int((c - a^2*c*x^2)^(5/2)/asin(a*x)^(1/2), x)

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