Optimal. Leaf size=98 \[ -\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {ArcSin}(a x)}}-\frac {2 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4751, 4731,
4491, 12, 3386, 3432} \begin {gather*} -\frac {2 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {ArcSin}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4731
Rule 4751
Rubi steps
\begin {align*} \int \frac {\sqrt {c-a^2 c x^2}}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (4 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (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} \sqrt {c-a^2 c x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (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 )}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\left (4 \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 )}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {2 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 83, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {c \left (1-a^2 x^2\right )} \left (1+\cos (2 \text {ArcSin}(a x))+2 \sqrt {\pi } \sqrt {\text {ArcSin}(a x)} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )\right )}{a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\arcsin \left (a x \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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