Optimal. Leaf size=245 \[ -\frac{3 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{128 b c^4}-\frac{\sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^4}+\frac{3 \sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4}+\frac{\sin \left (\frac{9 a}{b}\right ) \text{CosIntegral}\left (\frac{9 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{128 b c^4}+\frac{\cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^4}-\frac{3 \cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4}-\frac{\cos \left (\frac{9 a}{b}\right ) \text{Si}\left (\frac{9 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4} \]
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Rubi [A] time = 0.511682, 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} \[ -\frac{3 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b c^4}-\frac{\sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b c^4}+\frac{3 \sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b c^4}+\frac{\sin \left (\frac{9 a}{b}\right ) \text{CosIntegral}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b c^4}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b c^4}+\frac{\cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b c^4}-\frac{3 \cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b c^4}-\frac{\cos \left (\frac{9 a}{b}\right ) \text{Si}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b c^4} \]
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 )^{5/2}}{a+b \sin ^{-1}(c x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^6(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)}{128 (a+b x)}+\frac{\sin (3 x)}{32 (a+b x)}-\frac{3 \sin (7 x)}{256 (a+b x)}-\frac{\sin (9 x)}{256 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (9 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 c^4}+\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 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 )}{128 c^4}+\frac{\cos \left (\frac{3 a}{b}\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 )}{32 c^4}-\frac{\left (3 \cos \left (\frac{7 a}{b}\right )\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 )}{256 c^4}-\frac{\cos \left (\frac{9 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 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 )}{128 c^4}-\frac{\sin \left (\frac{3 a}{b}\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 )}{32 c^4}+\frac{\left (3 \sin \left (\frac{7 a}{b}\right )\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 )}{256 c^4}+\frac{\sin \left (\frac{9 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}\\ &=-\frac{3 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{128 b c^4}-\frac{\text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac{3 a}{b}\right )}{32 b c^4}+\frac{3 \text{Ci}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right ) \sin \left (\frac{7 a}{b}\right )}{256 b c^4}+\frac{\text{Ci}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right ) \sin \left (\frac{9 a}{b}\right )}{256 b c^4}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b c^4}+\frac{\cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b c^4}-\frac{3 \cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b c^4}-\frac{\cos \left (\frac{9 a}{b}\right ) \text{Si}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b c^4}\\ \end{align*}
Mathematica [A] time = 1.13601, size = 180, normalized size = 0.73 \[ \frac{-6 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-8 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 \sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac{9 a}{b}\right ) \text{CosIntegral}\left (9 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+6 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+8 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-3 \cos \left (\frac{7 a}{b}\right ) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-\cos \left (\frac{9 a}{b}\right ) \text{Si}\left (9 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{256 b c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 185, normalized size = 0.8 \begin{align*} -{\frac{1}{256\,{c}^{4}b} \left ( 3\,{\it Si} \left ( 7\,\arcsin \left ( cx \right ) +7\,{\frac{a}{b}} \right ) \cos \left ( 7\,{\frac{a}{b}} \right ) -3\,{\it Ci} \left ( 7\,\arcsin \left ( cx \right ) +7\,{\frac{a}{b}} \right ) \sin \left ( 7\,{\frac{a}{b}} \right ) -{\it Ci} \left ( 9\,\arcsin \left ( cx \right ) +9\,{\frac{a}{b}} \right ) \sin \left ( 9\,{\frac{a}{b}} \right ) -8\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) +8\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) +6\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) +{\it Si} \left ( 9\,\arcsin \left ( cx \right ) +9\,{\frac{a}{b}} \right ) \cos \left ( 9\,{\frac{a}{b}} \right ) -6\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\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{5}{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^{4} x^{7} - 2 \, 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.45036, size = 1007, normalized size = 4.11 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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