Optimal. Leaf size=72 \[ \sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )+\sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )-\sqrt{x+1} \cos (x) \]
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Rubi [A] time = 0.122283, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6129, 3296, 3306, 3305, 3351, 3304, 3352} \[ \sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )+\sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )-\sqrt{x+1} \cos (x) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} \sqrt{1-x} \sin (x) \, dx &=\int \sqrt{1+x} \sin (x) \, dx\\ &=-\sqrt{1+x} \cos (x)+\frac{1}{2} \int \frac{\cos (x)}{\sqrt{1+x}} \, dx\\ &=-\sqrt{1+x} \cos (x)+\frac{1}{2} \cos (1) \int \frac{\cos (1+x)}{\sqrt{1+x}} \, dx+\frac{1}{2} \sin (1) \int \frac{\sin (1+x)}{\sqrt{1+x}} \, dx\\ &=-\sqrt{1+x} \cos (x)+\cos (1) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1+x}\right )+\sin (1) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1+x}\right )\\ &=-\sqrt{1+x} \cos (x)+\sqrt{\frac{\pi }{2}} \cos (1) C\left (\sqrt{\frac{2}{\pi }} \sqrt{1+x}\right )+\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{1+x}\right ) \sin (1)\\ \end{align*}
Mathematica [C] time = 0.0235258, size = 77, normalized size = 1.07 \[ -\frac{e^{-i} \sqrt{x+1} \text{Gamma}\left (\frac{3}{2},-i (x+1)\right )}{2 \sqrt{-i (x+1)}}-\frac{e^i \sqrt{x+1} \text{Gamma}\left (\frac{3}{2},i (x+1)\right )}{2 \sqrt{i (x+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.302, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+x \right ) \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 = 1.29654, size = 672, normalized size = 9.33 \begin{align*} \text{result too large to display} \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 - 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{1 - x} \left (x + 1\right ) \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 = 1.1806, size = 89, normalized size = 1.24 \begin{align*} \left (\frac{1}{8} i - \frac{1}{8}\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}{8} i + \frac{1}{8}\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 )} - \frac{1}{2} \, \sqrt{x + 1} e^{\left (i \, x\right )} - \frac{1}{2} \, \sqrt{x + 1} e^{\left (-i \, x\right )} - 0.339605729125 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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