Optimal. Leaf size=144 \[ \frac{\cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b c}+\frac{\cos \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 b c}+\frac{\sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b c}+\frac{\sin \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 b c}+\frac{3 \log \left (a+b \sin ^{-1}(c x)\right )}{8 b c} \]
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Rubi [A] time = 0.242156, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4661, 3312, 3303, 3299, 3302} \[ \frac{\cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{2 b c}+\frac{\cos \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b c}+\frac{\sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{2 b c}+\frac{\sin \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b c}+\frac{3 \log \left (a+b \sin ^{-1}(c x)\right )}{8 b c} \]
Antiderivative was successfully verified.
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Rule 4661
Rule 3312
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{3/2}}{a+b \sin ^{-1}(c x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 (a+b x)}+\frac{\cos (2 x)}{2 (a+b x)}+\frac{\cos (4 x)}{8 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c}\\ &=\frac{3 \log \left (a+b \sin ^{-1}(c x)\right )}{8 b c}+\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}\\ &=\frac{3 \log \left (a+b \sin ^{-1}(c x)\right )}{8 b c}+\frac{\cos \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}+\frac{\cos \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c}+\frac{\sin \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}+\frac{\sin \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c}\\ &=\frac{\cos \left (\frac{2 a}{b}\right ) \text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{2 b c}+\frac{\cos \left (\frac{4 a}{b}\right ) \text{Ci}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b c}+\frac{3 \log \left (a+b \sin ^{-1}(c x)\right )}{8 b c}+\frac{\sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{2 b c}+\frac{\sin \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b c}\\ \end{align*}
Mathematica [A] time = 0.323578, size = 121, normalized size = 0.84 \[ \frac{4 \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\cos \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+4 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac{4 a}{b}\right ) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+4 \log \left (a+b \sin ^{-1}(c x)\right )-\log \left (8 \left (a+b \sin ^{-1}(c x)\right )\right )}{8 b c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 135, normalized size = 0.9 \begin{align*}{\frac{1}{8\,cb}{\it Si} \left ( 4\,\arcsin \left ( cx \right ) +4\,{\frac{a}{b}} \right ) \sin \left ( 4\,{\frac{a}{b}} \right ) }+{\frac{1}{8\,cb}{\it Ci} \left ( 4\,\arcsin \left ( cx \right ) +4\,{\frac{a}{b}} \right ) \cos \left ( 4\,{\frac{a}{b}} \right ) }+{\frac{1}{2\,cb}{\it Si} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) }+{\frac{1}{2\,cb}{\it Ci} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) }+{\frac{3\,\ln \left ( a+b\arcsin \left ( cx \right ) \right ) }{8\,cb}} \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*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{b \arcsin \left (c x\right ) + a}\,{d x} \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{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{b \arcsin \left (c x\right ) + a}, 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*} \int \frac{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{a + b \operatorname{asin}{\left (c x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40511, size = 340, normalized size = 2.36 \begin{align*} \frac{\cos \left (\frac{a}{b}\right )^{4} \operatorname{Ci}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c} + \frac{\cos \left (\frac{a}{b}\right )^{3} \sin \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c} - \frac{\cos \left (\frac{a}{b}\right )^{2} \operatorname{Ci}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c} + \frac{\cos \left (\frac{a}{b}\right )^{2} \operatorname{Ci}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c} - \frac{\cos \left (\frac{a}{b}\right ) \sin \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c} + \frac{\cos \left (\frac{a}{b}\right ) \sin \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c} + \frac{\operatorname{Ci}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{8 \, b c} - \frac{\operatorname{Ci}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c} + \frac{3 \, \log \left (b \arcsin \left (c x\right ) + a\right )}{8 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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