Optimal. Leaf size=62 \[ \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \]
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Rubi [A] time = 0.110165, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6129, 3306, 3305, 3351, 3304, 3352} \[ \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(x)} \sin (x)}{\sqrt{1+x}} \, dx &=\int \frac{\sin (x)}{\sqrt{1-x}} \, dx\\ &=-\left (\cos (1) \int \frac{\sin (1-x)}{\sqrt{1-x}} \, dx\right )+\sin (1) \int \frac{\cos (1-x)}{\sqrt{1-x}} \, dx\\ &=(2 \cos (1)) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-(2 \sin (1)) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=\sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)\\ \end{align*}
Mathematica [C] time = 0.0274356, size = 70, normalized size = 1.13 \[ -\frac{e^{-i} \left (e^{2 i} \sqrt{-i (x-1)} \text{Gamma}\left (\frac{1}{2},-i (x-1)\right )+\sqrt{i (x-1)} \text{Gamma}\left (\frac{1}{2},i (x-1)\right )\right )}{2 \sqrt{1-x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.281, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( x \right ) \sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.1369, size = 227, normalized size = 3.66 \begin{align*} -\frac{{\left ({\left ({\left (i \, \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{i \, x - i}\right ) - 1\right )} - i \, \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-i \, x + i}\right ) - 1\right )}\right )} \cos \left (1\right ) +{\left (\sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{i \, x - i}\right ) - 1\right )} + \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-i \, x + i}\right ) - 1\right )}\right )} \sin \left (1\right )\right )} \cos \left (\frac{1}{2} \, \arctan \left (x - 1, 0\right )\right ) +{\left ({\left (\sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{i \, x - i}\right ) - 1\right )} + \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-i \, x + i}\right ) - 1\right )}\right )} \cos \left (1\right ) +{\left (-i \, \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{i \, x - i}\right ) - 1\right )} + i \, \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-i \, x + i}\right ) - 1\right )}\right )} \sin \left (1\right )\right )} \sin \left (\frac{1}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} \sqrt{-x + 1}}{2 \, \sqrt{{\left | x - 1 \right |}}} \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{\sqrt{-x^{2} + 1} \sqrt{x + 1} \sin \left (x\right )}{x^{2} - 1}, x\right ) \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*} \int \frac{\sqrt{x + 1} \sin{\left (x \right )}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.20691, size = 65, normalized size = 1.05 \begin{align*} -\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{-x + 1}\right ) e^{i} + \left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{-x + 1}\right ) e^{\left (-i\right )} + 0.82661965415 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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