Optimal. Leaf size=217 \[ \frac{15 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c}+\frac{3 \sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2 c}+\frac{3 \sin \left (\frac{6 a}{b}\right ) \text{CosIntegral}\left (\frac{6 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c}-\frac{15 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c}-\frac{3 \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2 c}-\frac{3 \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (\frac{6 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c}-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.399717, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {4659, 4723, 4406, 3303, 3299, 3302} \[ \frac{15 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c}+\frac{3 \sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{4 b^2 c}+\frac{3 \sin \left (\frac{6 a}{b}\right ) \text{CosIntegral}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c}-\frac{15 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c}-\frac{3 \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{4 b^2 c}-\frac{3 \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c}-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 4659
Rule 4723
Rule 4406
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{5/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{(6 c) \int \frac{x \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{6 \operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{6 \operatorname{Subst}\left (\int \left (\frac{5 \sin (2 x)}{32 (a+b x)}+\frac{\sin (4 x)}{8 (a+b x)}+\frac{\sin (6 x)}{32 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}-\frac{15 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c}\\ &=-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\left (15 \cos \left (\frac{2 a}{b}\right )\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 )}{16 b c}-\frac{\left (3 \cos \left (\frac{4 a}{b}\right )\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 )}{4 b c}-\frac{\left (3 \cos \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c}+\frac{\left (15 \sin \left (\frac{2 a}{b}\right )\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 )}{16 b c}+\frac{\left (3 \sin \left (\frac{4 a}{b}\right )\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 )}{4 b c}+\frac{\left (3 \sin \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c}\\ &=-\frac{\left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{15 \text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right ) \sin \left (\frac{2 a}{b}\right )}{16 b^2 c}+\frac{3 \text{Ci}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right ) \sin \left (\frac{4 a}{b}\right )}{4 b^2 c}+\frac{3 \text{Ci}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right ) \sin \left (\frac{6 a}{b}\right )}{16 b^2 c}-\frac{15 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c}-\frac{3 \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{4 b^2 c}-\frac{3 \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.82457, size = 311, normalized size = 1.43 \[ -\frac{-15 \sin \left (\frac{2 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-12 \sin \left (\frac{4 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-3 a \sin \left (\frac{6 a}{b}\right ) \text{CosIntegral}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-3 b \sin \left (\frac{6 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+15 a \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+15 b \cos \left (\frac{2 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+12 a \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+12 b \cos \left (\frac{4 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 a \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 b \cos \left (\frac{6 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-16 b c^6 x^6+48 b c^4 x^4-48 b c^2 x^2+16 b}{16 b^2 c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 364, normalized size = 1.7 \begin{align*} -{\frac{1}{32\,c \left ( a+b\arcsin \left ( cx \right ) \right ){b}^{2}} \left ( 6\,\arcsin \left ( cx \right ){\it Si} \left ( 6\,\arcsin \left ( cx \right ) +6\,{\frac{a}{b}} \right ) \cos \left ( 6\,{\frac{a}{b}} \right ) b-6\,\arcsin \left ( cx \right ){\it Ci} \left ( 6\,\arcsin \left ( cx \right ) +6\,{\frac{a}{b}} \right ) \sin \left ( 6\,{\frac{a}{b}} \right ) b+24\,\arcsin \left ( cx \right ){\it Si} \left ( 4\,\arcsin \left ( cx \right ) +4\,{\frac{a}{b}} \right ) \cos \left ( 4\,{\frac{a}{b}} \right ) b-24\,\arcsin \left ( cx \right ){\it Ci} \left ( 4\,\arcsin \left ( cx \right ) +4\,{\frac{a}{b}} \right ) \sin \left ( 4\,{\frac{a}{b}} \right ) b+30\,\arcsin \left ( cx \right ){\it Si} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) b-30\,\arcsin \left ( cx \right ){\it Ci} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) b+6\,{\it Si} \left ( 6\,\arcsin \left ( cx \right ) +6\,{\frac{a}{b}} \right ) \cos \left ( 6\,{\frac{a}{b}} \right ) a-6\,{\it Ci} \left ( 6\,\arcsin \left ( cx \right ) +6\,{\frac{a}{b}} \right ) \sin \left ( 6\,{\frac{a}{b}} \right ) a+24\,{\it Si} \left ( 4\,\arcsin \left ( cx \right ) +4\,{\frac{a}{b}} \right ) \cos \left ( 4\,{\frac{a}{b}} \right ) a-24\,{\it Ci} \left ( 4\,\arcsin \left ( cx \right ) +4\,{\frac{a}{b}} \right ) \sin \left ( 4\,{\frac{a}{b}} \right ) a+30\,{\it Si} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) a-30\,{\it Ci} \left ( 2\,\arcsin \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) a+\cos \left ( 6\,\arcsin \left ( cx \right ) \right ) b+6\,\cos \left ( 4\,\arcsin \left ( cx \right ) \right ) b+15\,\cos \left ( 2\,\arcsin \left ( cx \right ) \right ) b+10\,b \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{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 6 \,{\left (b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c\right )} \int \frac{c^{5} x^{5} - 2 \, c^{3} x^{3} + c x}{b^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b}\,{d x} - 1}{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^{4} x^{4} - 2 \, c^{2} x^{2} + 1\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 = 1.59521, size = 1882, normalized size = 8.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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