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

Optimal. Leaf size=103 \[ \frac {4 a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\sin ^{-1}(a x)}},x\right )}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}} \]

[Out]

-2*(-a^2*x^2+1)^(1/2)/a/(-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(1/2)+4*a*(-a^2*x^2+1)^(1/2)*Unintegrable(x/(-a^2*x^2
+1)^2/arcsin(a*x)^(1/2),x)/c/(-a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2)),x]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(a*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]]) + (4*a*Sqrt[1 - a^2*x^2]*Defer[Int][x/((1 -
a^2*x^2)^2*Sqrt[ArcSin[a*x]]), x])/(c*Sqrt[c - a^2*c*x^2])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.94, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2)),x]

[Out]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2)), x]

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(3/2)*arcsin(a*x)^(3/2)), x)

________________________________________________________________________________________

maple [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsin \left (a x \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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(1/(-a^2*c*x^2+c)^(3/2)/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.

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{3/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________