Optimal. Leaf size=170 \[ \frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{4 a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.154473, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4663, 4661, 3312, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{4 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4663
Rule 4661
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\left (c-a^2 c x^2\right )^{3/2}}{\sqrt{\sin ^{-1}(a x)}} \, dx &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{\sqrt{\sin ^{-1}(a x)}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{4 a \sqrt{1-a^2 x^2}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{4 a \sqrt{1-a^2 x^2}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a \sqrt{1-a^2 x^2}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a \sqrt{1-a^2 x^2}}\\ &=\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{4 a \sqrt{1-a^2 x^2}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.305299, size = 182, normalized size = 1.07 \[ \frac{c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \left (-4 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )-4 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )+24 \sqrt{\sin ^{-1}(a x)^2}\right )}{32 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\sqrt{\operatorname{asin}{\left (a x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}{\sqrt{\arcsin \left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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