Optimal. Leaf size=141 \[ -2 \sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+2 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{1-x} \cos (x) \]
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Rubi [A] time = 0.170036, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6129, 6742, 3353, 3352, 3351, 3385, 3354} \[ -2 \sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+2 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{1-x} \cos (x) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 6742
Rule 3353
Rule 3352
Rule 3351
Rule 3385
Rule 3354
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} \sqrt{1+x} \sin (x) \, dx &=\int \frac{(1+x) \sin (x)}{\sqrt{1-x}} \, dx\\ &=2 \operatorname{Subst}\left (\int \left (-2+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-2 \sin \left (1-x^2\right )+x^2 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt{1-x}\right )\\ &=2 \operatorname{Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )-4 \operatorname{Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=\sqrt{1-x} \cos (x)+(4 \cos (1)) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-(4 \sin (1)) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-\operatorname{Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=\sqrt{1-x} \cos (x)+2 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-2 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-\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 )+2 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-2 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)\\ \end{align*}
Mathematica [C] time = 7.98664, size = 134, normalized size = 0.95 \[ \left (\frac{1}{8}+\frac{i}{8}\right ) \left (\frac{e^{-i x} \sqrt{1-x^2} \left ((4+i) \sqrt{2 \pi } e^{i (x+1)} \text{Erfi}\left (\frac{(1+i) \sqrt{x-1}}{\sqrt{2}}\right )+(2-2 i) \sqrt{x-1} \left (1+e^{2 i x}\right )\right )}{\sqrt{x-1} \sqrt{x+1}}-(4-i) e^{-i} \sqrt{2 \pi } \text{Erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{2-2 x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.277, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( x \right ) \left ( 1+x \right ) ^{{\frac{3}{2}}}{\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 = 1.21529, size = 471, normalized size = 3.34 \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(-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 [C] time = 1.224, size = 100, normalized size = 0.71 \begin{align*} -\left (\frac{5}{8} i + \frac{3}{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{5}{8} i - \frac{3}{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 )} + 1.99284503743 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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