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

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

[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 = 0.19, 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 = {4659, 4723, 4406, 3305, 3351} \[ -\frac {3 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a \sqrt {1-a^2 x^2}}-\frac {\sqrt {3 \pi } c^2 \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {15 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\sin ^{-1}(a x)}} \]

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 3305

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 3351

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

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]

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 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])

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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]  time = 1.26, size = 404, normalized size = 1.70 \[ -\frac {c^2 \sqrt {c-a^2 c x^2} e^{-6 i \sin ^{-1}(a x)} \left (64 \sqrt {\pi } e^{6 i \sin ^{-1}(a x)} \sqrt {\sin ^{-1}(a x)} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )+6 e^{2 i \sin ^{-1}(a x)}+15 e^{4 i \sin ^{-1}(a x)}+20 e^{6 i \sin ^{-1}(a x)}+15 e^{8 i \sin ^{-1}(a x)}+6 e^{10 i \sin ^{-1}(a x)}+e^{12 i \sin ^{-1}(a x)}+\sqrt {2} e^{6 i \sin ^{-1}(a x)} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt {2} e^{6 i \sin ^{-1}(a x)} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )-12 e^{6 i \sin ^{-1}(a x)} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )-12 e^{6 i \sin ^{-1}(a x)} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )-\sqrt {6} e^{6 i \sin ^{-1}(a x)} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 i \sin ^{-1}(a x)\right )-\sqrt {6} e^{6 i \sin ^{-1}(a x)} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},6 i \sin ^{-1}(a x)\right )+1\right )}{32 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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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maple [F]  time = 0.38, 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 \[ \text {Exception raised: RuntimeError} \]

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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