Optimal. Leaf size=253 \[ \frac{3 \sqrt{\frac{\pi }{2}} d \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}+\frac{\sqrt{\frac{3 \pi }{2}} d \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{3 \sqrt{\frac{\pi }{2}} d \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{\sqrt{\frac{3 \pi }{2}} d \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
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Rubi [A] time = 0.591869, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4659, 4723, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{3 \sqrt{\frac{\pi }{2}} d \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}+\frac{\sqrt{\frac{3 \pi }{2}} d \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{3 \sqrt{\frac{\pi }{2}} d \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{\sqrt{\frac{3 \pi }{2}} d \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
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{d-c^2 d x^2}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(6 c d) \int \frac{x \sqrt{1-c^2 x^2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(6 d) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(6 d) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{a+b x}}+\frac{\sin (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (3 d \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 )}{2 b c}-\frac{\left (3 d \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 )}{2 b c}+\frac{\left (3 d \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 )}{2 b c}+\frac{\left (3 d \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 )}{2 b c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{\left (3 d \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 )}{b^2 c}-\frac{\left (3 d \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 )}{b^2 c}+\frac{\left (3 d \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 )}{b^2 c}+\frac{\left (3 d \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 )}{b^2 c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{3 d \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 )}{b^{3/2} c}-\frac{d \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 )}{b^{3/2} c}+\frac{3 d \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 )}{b^{3/2} c}+\frac{d \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 )}{b^{3/2} c}\\ \end{align*}
Mathematica [C] time = 1.0961, size = 348, normalized size = 1.38 \[ \frac{d e^{-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}} \left (3 e^{\frac{2 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3 e^{\frac{4 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{3} e^{3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{3} e^{3 i \left (\frac{2 a}{b}+\sin ^{-1}(c x)\right )} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-3 e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c x)}-3 e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c x)}-e^{\frac{3 i \left (a+2 b \sin ^{-1}(c x)\right )}{b}}-e^{\frac{3 i a}{b}}\right )}{4 b c \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.074, size = 297, normalized size = 1.2 \begin{align*} -{\frac{d}{2\,bc} \left ( \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{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\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{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) +3\,\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{{b}^{-1}}\sqrt{\pi }b}} \right ) -3\,\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{{b}^{-1}}\sqrt{\pi }b}} \right ) +3\,\cos \left ({\frac{a+b\arcsin \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) +\cos \left ( 3\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-3\,{\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{c^{2} d x^{2} - d}{{\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 \left (\int \frac{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{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{c^{2} d x^{2} - d}{{\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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