Optimal. Leaf size=58 \[ \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {2 \pi } \sin (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3306, 3305, 3351, 3304, 3352} \[ \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {2 \pi } \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rubi steps
\begin {align*} \int \frac {\sin (x)}{\sqrt {1+x}} \, dx &=\cos (1) \int \frac {\sin (1+x)}{\sqrt {1+x}} \, dx-\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 = 0.02, size = 68, normalized size = 1.17 \[ -\frac {e^{-i} \left (\sqrt {-i (x+1)} \operatorname {Gamma}\left (\frac {1}{2},-i (x+1)\right )+e^{2 i} \sqrt {i (x+1)} \operatorname {Gamma}\left (\frac {1}{2},i (x+1)\right )\right )}{2 \sqrt {x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 46, normalized size = 0.79 \[ \sqrt {2} \sqrt {\pi } \cos \relax (1) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {x + 1}}{\sqrt {\pi }}\right ) - \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} \sqrt {x + 1}}{\sqrt {\pi }}\right ) \sin \relax (1) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.18, size = 43, normalized size = 0.74 \[ -\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{i} + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{\left (-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 42, normalized size = 0.72 \[ \sqrt {2}\, \sqrt {\pi }\, \left (-\sin \relax (1) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {x +1}}{\sqrt {\pi }}\right )+\cos \relax (1) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {x +1}}{\sqrt {\pi }}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.38, size = 112, normalized size = 1.93 \[ \frac {1}{8} \, \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \sqrt {2} \cos \relax (1) + \left (i - 1\right ) \, \sqrt {2} \sin \relax (1)\right )} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \cos \relax (1) + \left (i + 1\right ) \, \sqrt {2} \sin \relax (1)\right )} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) + {\left (-\left (i - 1\right ) \, \sqrt {2} \cos \relax (1) - \left (i + 1\right ) \, \sqrt {2} \sin \relax (1)\right )} \operatorname {erf}\left (\sqrt {-i} \sqrt {x + 1}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \cos \relax (1) + \left (i - 1\right ) \, \sqrt {2} \sin \relax (1)\right )} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} \sqrt {x + 1}\right )\right )} \]
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 \[ \int \frac {\sin \relax (x)}{\sqrt {x+1}} \,d x \]
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 \[ \int \frac {\sin {\relax (x )}}{\sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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