Optimal. Leaf size=62 \[ \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {2 \pi } \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 6129
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*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 1.13 \[ -\frac {e^{-i} \left (e^{2 i} \sqrt {-i (x-1)} \Gamma \left (\frac {1}{2},-i (x-1)\right )+\sqrt {i (x-1)} \Gamma \left (\frac {1}{2},i (x-1)\right )\right )}{2 \sqrt {1-x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} \sqrt {x + 1} \sin \relax (x)}{x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.18, size = 48, normalized size = 0.77 \[ -\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{i} + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{\left (-i\right )} + 0.826619654151000 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1+x}\, \sin \relax (x )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 168, normalized size = 2.71 \[ -\frac {{\left ({\left ({\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \cos \relax (1) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \sin \relax (1)\right )} \cos \left (\frac {1}{2} \, \arctan \left (x - 1, 0\right )\right ) + {\left ({\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \cos \relax (1) + {\left (-i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \sin \relax (1)\right )} \sin \left (\frac {1}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} \sqrt {-x + 1}}{2 \, \sqrt {{\left | x - 1 \right |}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \relax (x)\,\sqrt {x+1}}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + 1} \sin {\relax (x )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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