Optimal. Leaf size=245 \[ -\frac{3 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{64 b c^4}-\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^4}+\frac{\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^4}+\frac{\sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^4}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{64 b c^4}+\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^4}-\frac{\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^4}-\frac{\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^4} \]
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Rubi [A] time = 0.499066, antiderivative size = 241, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4723, 4406, 3303, 3299, 3302} \[ -\frac{3 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^4}+\frac{\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^4}+\frac{\sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^4}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^4} \]
Antiderivative was successfully verified.
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Rule 4723
Rule 4406
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^3 \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) \sin ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{64 (a+b x)}+\frac{3 \sin (3 x)}{64 (a+b x)}-\frac{\sin (5 x)}{64 (a+b x)}-\frac{\sin (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}-\frac{\operatorname{Subst}\left (\int \frac{\sin (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}\\ &=\frac{\left (3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}+\frac{\left (3 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}-\frac{\cos \left (\frac{5 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}-\frac{\cos \left (\frac{7 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}-\frac{\left (3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}-\frac{\left (3 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}+\frac{\sin \left (\frac{5 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}+\frac{\sin \left (\frac{7 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^4}\\ &=-\frac{3 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{64 b c^4}-\frac{3 \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac{3 a}{b}\right )}{64 b c^4}+\frac{\text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right ) \sin \left (\frac{5 a}{b}\right )}{64 b c^4}+\frac{\text{Ci}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right ) \sin \left (\frac{7 a}{b}\right )}{64 b c^4}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^4}\\ \end{align*}
Mathematica [A] time = 0.752081, size = 179, normalized size = 0.73 \[ \frac{-3 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-3 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+3 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{64 b c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 184, normalized size = 0.8 \begin{align*}{\frac{1}{64\,{c}^{4}b} \left ( 3\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) +{\it Ci} \left ( 7\,\arcsin \left ( cx \right ) +7\,{\frac{a}{b}} \right ) \sin \left ( 7\,{\frac{a}{b}} \right ) -3\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) -{\it Si} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \cos \left ( 5\,{\frac{a}{b}} \right ) +{\it Ci} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \sin \left ( 5\,{\frac{a}{b}} \right ) +3\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) -3\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) -{\it Si} \left ( 7\,\arcsin \left ( cx \right ) +7\,{\frac{a}{b}} \right ) \cos \left ( 7\,{\frac{a}{b}} \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*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{3}}{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^{5} - x^{3}\right )} \sqrt{-c^{2} x^{2} + 1}}{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{x^{3} \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 [B] time = 1.39014, size = 829, normalized size = 3.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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