Optimal. Leaf size=390 \[ \frac{5 \sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c}+\frac{5 \sqrt{\frac{3 \pi }{2}} d^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}+\frac{\sqrt{\frac{5 \pi }{2}} d^2 \sin \left (\frac{5 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{5 \sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c}-\frac{5 \sqrt{\frac{3 \pi }{2}} d^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{\sqrt{\frac{5 \pi }{2}} d^2 \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
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Rubi [A] time = 0.816396, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {4659, 4723, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{5 \sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c}+\frac{5 \sqrt{\frac{3 \pi }{2}} d^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}+\frac{\sqrt{\frac{5 \pi }{2}} d^2 \sin \left (\frac{5 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{5 \sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c}-\frac{5 \sqrt{\frac{3 \pi }{2}} d^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{\sqrt{\frac{5 \pi }{2}} d^2 \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 4659
Rule 4723
Rule 4406
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (10 c d^2\right ) \int \frac{x \left (1-c^2 x^2\right )^{3/2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (10 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (10 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{8 \sqrt{a+b x}}+\frac{3 \sin (3 x)}{16 \sqrt{a+b x}}+\frac{\sin (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}-\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (5 d^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}-\frac{\left (15 d^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}-\frac{\left (5 d^2 \cos \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{5 a}{b}+5 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}+\frac{\left (5 d^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}+\frac{\left (15 d^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}+\frac{\left (5 d^2 \sin \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{5 a}{b}+5 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (5 d^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{2 b^2 c}-\frac{\left (15 d^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}-\frac{\left (5 d^2 \cos \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{5 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}+\frac{\left (5 d^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{2 b^2 c}+\frac{\left (15 d^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}+\frac{\left (5 d^2 \sin \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{5 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}\\ &=-\frac{2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{5 d^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c}-\frac{5 d^2 \sqrt{\frac{3 \pi }{2}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}-\frac{d^2 \sqrt{\frac{5 \pi }{2}} \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c}+\frac{5 d^2 \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 b^{3/2} c}+\frac{5 d^2 \sqrt{\frac{3 \pi }{2}} C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{4 b^{3/2} c}+\frac{d^2 \sqrt{\frac{5 \pi }{2}} C\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{5 a}{b}\right )}{4 b^{3/2} c}\\ \end{align*}
Mathematica [C] time = 2.40087, size = 522, normalized size = 1.34 \[ \frac{d^2 e^{-\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}} \left (10 e^{\frac{4 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+10 e^{\frac{6 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+5 \sqrt{3} e^{\frac{2 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+5 \sqrt{3} e^{\frac{8 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{5} e^{5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{5} e^{\frac{5 i \left (2 a+b \sin ^{-1}(c x)\right )}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-5 e^{\frac{5 i a}{b}+2 i \sin ^{-1}(c x)}-10 e^{\frac{5 i a}{b}+4 i \sin ^{-1}(c x)}-10 e^{\frac{5 i a}{b}+6 i \sin ^{-1}(c x)}-5 e^{\frac{5 i a}{b}+8 i \sin ^{-1}(c x)}-e^{\frac{5 i \left (a+2 b \sin ^{-1}(c x)\right )}{b}}-e^{\frac{5 i a}{b}}\right )}{16 b c \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.087, size = 446, normalized size = 1.1 \begin{align*} -{\frac{{d}^{2}}{8\,bc} \left ( 5\,\sqrt{3}\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -5\,\sqrt{3}\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\sqrt{5}\cos \left ( 5\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{5}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\sqrt{5}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{5}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sin \left ( 5\,{\frac{a}{b}} \right ) +10\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -10\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +10\,\cos \left ({\frac{a+b\arcsin \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) +5\,\cos \left ( 3\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-3\,{\frac{a}{b}} \right ) +\cos \left ( 5\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-5\,{\frac{a}{b}} \right ) \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \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} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} d^{2} \left (\int - \frac{2 c^{2} x^{2}}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int \frac{c^{4} x^{4}}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int \frac{1}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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