Optimal. Leaf size=278 \[ \frac{3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{64 b^2 c^4}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{7 \cos \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}+\frac{3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{64 b^2 c^4}+\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{7 \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.889903, antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 28, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4721, 4723, 4406, 3303, 3299, 3302} \[ \frac{3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{7 \cos \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{7 \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 4721
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}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \int \frac{x^2 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac{(7 c) \int \frac{x^4 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{7 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 (a+b x)}-\frac{\cos (3 x)}{16 (a+b x)}-\frac{\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{7 \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{64 (a+b x)}-\frac{3 \cos (3 x)}{64 (a+b x)}-\frac{\cos (5 x)}{64 (a+b x)}+\frac{\cos (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{7 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{7 \operatorname{Subst}\left (\int \frac{\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac{21 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{21 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\left (21 \cos \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 b c^4}+\frac{\left (3 \cos \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 )}{8 b c^4}-\frac{\left (3 \cos \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 )}{16 b c^4}+\frac{\left (21 \cos \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 b c^4}+\frac{\left (7 \cos \left (\frac{5 a}{b}\right )\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 b c^4}-\frac{\left (3 \cos \left (\frac{5 a}{b}\right )\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 )}{16 b c^4}-\frac{\left (7 \cos \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 )}{64 b c^4}-\frac{\left (21 \sin \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 b c^4}+\frac{\left (3 \sin \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 )}{8 b c^4}-\frac{\left (3 \sin \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 )}{16 b c^4}+\frac{\left (21 \sin \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 b c^4}+\frac{\left (7 \sin \left (\frac{5 a}{b}\right )\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 b c^4}-\frac{\left (3 \sin \left (\frac{5 a}{b}\right )\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 )}{16 b c^4}-\frac{\left (7 \sin \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 )}{64 b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{7 \cos \left (\frac{7 a}{b}\right ) \text{Ci}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac{7 \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}\\ \end{align*}
Mathematica [A] time = 1.05366, size = 399, normalized size = 1.44 \[ -\frac{-3 \cos \left (\frac{a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-9 \cos \left (\frac{3 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \cos \left (\frac{5 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+7 a \cos \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+7 b \cos \left (\frac{7 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-3 a \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-3 b \sin \left (\frac{a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-9 a \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-9 b \sin \left (\frac{3 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \sin \left (\frac{5 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+7 a \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+7 b \sin \left (\frac{7 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+64 b c^7 x^7-128 b c^5 x^5+64 b c^3 x^3}{64 b^2 c^4 \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 455, normalized size = 1.6 \begin{align*} \text{result too large to display} \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{c^{4} x^{7} - 2 \, c^{2} x^{5} + x^{3} - \frac{{\left (7 \, c^{4} \int \frac{x^{6}}{b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a}\,{d x} - 10 \, c^{2} \int \frac{x^{4}}{b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a}\,{d x} + 3 \, \int \frac{x^{2}}{b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a}\,{d x}\right )}{\left (b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c\right )}}{b c}}{b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c} \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^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, 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 = 1.74399, size = 2788, normalized size = 10.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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