Optimal. Leaf size=55 \[ -\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-\frac{3 c \text{Si}\left (\sin ^{-1}(a x)\right )}{4 a}-\frac{3 c \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a} \]
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Rubi [A] time = 0.117651, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4659, 4723, 4406, 3299} \[ -\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-\frac{3 c \text{Si}\left (\sin ^{-1}(a x)\right )}{4 a}-\frac{3 c \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 4659
Rule 4723
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int \frac{c-a^2 c x^2}{\sin ^{-1}(a x)^2} \, dx &=-\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-(3 a c) \int \frac{x \sqrt{1-a^2 x^2}}{\sin ^{-1}(a x)} \, dx\\ &=-\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-\frac{(3 c) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a}\\ &=-\frac{c \left (1-a^2 x^2\right )^{3/2}}{a \sin ^{-1}(a x)}-\frac{3 c \text{Si}\left (\sin ^{-1}(a x)\right )}{4 a}-\frac{3 c \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.222179, size = 55, normalized size = 1. \[ -\frac{c \left (4 \left (1-a^2 x^2\right )^{3/2}+3 \sin ^{-1}(a x) \text{Si}\left (\sin ^{-1}(a x)\right )+3 \sin ^{-1}(a x) \text{Si}\left (3 \sin ^{-1}(a x)\right )\right )}{4 a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 59, normalized size = 1.1 \begin{align*} -{\frac{c}{4\,a\arcsin \left ( ax \right ) } \left ( 3\,{\it Si} \left ( \arcsin \left ( ax \right ) \right ) \arcsin \left ( ax \right ) +3\,{\it Si} \left ( 3\,\arcsin \left ( ax \right ) \right ) \arcsin \left ( ax \right ) +3\,\sqrt{-{a}^{2}{x}^{2}+1}+\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{3 \, a^{2} c \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right ) \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x}{\arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}\,{d x} -{\left (a^{2} c x^{2} - c\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{a \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}}, x\right ) \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*} - c \left (\int \frac{a^{2} x^{2}}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx + \int - \frac{1}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38418, size = 66, normalized size = 1.2 \begin{align*} -\frac{3 \, c \operatorname{Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a} - \frac{3 \, c \operatorname{Si}\left (\arcsin \left (a x\right )\right )}{4 \, a} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{a \arcsin \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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