Optimal. Leaf size=72 \[ \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {x+1} \cos (x) \]
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Rubi [A] time = 0.12, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 6129
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*}
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Mathematica [C] time = 0.02, size = 77, normalized size = 1.07 \[ -\frac {e^{-i} \sqrt {x+1} \Gamma \left (\frac {3}{2},-i (x+1)\right )}{2 \sqrt {-i (x+1)}}-\frac {e^i \sqrt {x+1} \Gamma \left (\frac {3}{2},i (x+1)\right )}{2 \sqrt {i (x+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} \sqrt {-x + 1} \sin \relax (x)}{x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.18, size = 66, normalized size = 0.92 \[ \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{i} - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{\left (-i\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (i \, x\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (-i \, x\right )} - 0.339605729125000 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+x \right ) \sqrt {1-x}\, \sin \relax (x )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.44, size = 498, normalized size = 6.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \relax (x)\,\sqrt {1-x}\,\left (x+1\right )}{\sqrt {1-x^2}} \,d x \]
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 \[ \int \frac {\sqrt {1 - x} \left (x + 1\right ) \sin {\relax (x )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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