Optimal. Leaf size=163 \[ \frac {3}{2} \sqrt {\frac {\pi }{2}} \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\frac {3}{2} \sqrt {x+1} \sin (x)+(x+1)^{3/2} (-\cos (x))+\sqrt {x+1} \cos (x) \]
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Rubi [A] time = 0.30, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6129, 6742, 3385, 3354, 3352, 3351, 3386, 3353} \[ \frac {3}{2} \sqrt {\frac {\pi }{2}} \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\frac {3}{2} \sqrt {x+1} \sin (x)+(x+1)^{3/2} (-\cos (x))+\sqrt {x+1} \cos (x) \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3354
Rule 3385
Rule 3386
Rule 6129
Rule 6742
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} \sqrt {1-x} x \sin (x) \, dx &=\int x \sqrt {1+x} \sin (x) \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \left (-x^2 \sin \left (1-x^2\right )+x^4 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt {1+x}\right )\right )\\ &=2 \operatorname {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int x^4 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=\sqrt {1+x} \cos (x)-(1+x)^{3/2} \cos (x)+3 \operatorname {Subst}\left (\int x^2 \cos \left (1-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)-(1+x)^{3/2} \cos (x)+\frac {3}{2} \sqrt {1+x} \sin (x)+\frac {3}{2} \operatorname {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-\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)-(1+x)^{3/2} \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)+\frac {3}{2} \sqrt {1+x} \sin (x)-\frac {1}{2} (3 \cos (1)) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} (3 \sin (1)) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=\sqrt {1+x} \cos (x)-(1+x)^{3/2} \cos (x)-\sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} 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)+\frac {3}{2} \sqrt {1+x} \sin (x)\\ \end {align*}
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Mathematica [C] time = 8.68, size = 168, normalized size = 1.03 \[ \frac {\left (\frac {1}{16}+\frac {i}{16}\right ) e^{-i (x+1)} \sqrt {1-x} \left (e^i \left ((3-2 i) \sqrt {2 \pi } e^{i (x+1)} \sqrt {-x-1} \text {erfi}\left (\frac {(1+i) \sqrt {-x-1}}{\sqrt {2}}\right )+(2+2 i) \left (e^{2 i x} (-3+2 i x)+2 i x+3\right ) (x+1)\right )-(3+2 i) \sqrt {2 \pi } e^{i x} \sqrt {-x-1} \text {erf}\left (\frac {(1+i) \sqrt {-x-1}}{\sqrt {2}}\right )\right )}{\sqrt {1-x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} x \sqrt {-x + 1} \sin \relax (x)}{x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.19, size = 108, normalized size = 0.66 \[ \left (\frac {1}{16} i + \frac {5}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{i} - \left (\frac {1}{16} i - \frac {5}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{\left (-i\right )} + \frac {1}{4} i \, {\left (2 i \, {\left (x + 1\right )}^{\frac {3}{2}} - \left (4 i + 3\right ) \, \sqrt {x + 1}\right )} e^{\left (i \, x\right )} + \frac {1}{4} i \, {\left (2 i \, {\left (x + 1\right )}^{\frac {3}{2}} - \left (4 i - 3\right ) \, \sqrt {x + 1}\right )} e^{\left (-i \, x\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (i \, x\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (-i \, x\right )} - 0.537182832596000 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+x \right ) \sqrt {1-x}\, x \sin \relax (x )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.50, size = 909, normalized size = 5.58 \[ \text {result too large to display} \]
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 \[ \int \frac {x\,\sin \relax (x)\,\sqrt {1-x}\,\left (x+1\right )}{\sqrt {1-x^2}} \,d x \]
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 \[ \int \frac {x \sqrt {1 - x} \left (x + 1\right ) \sin {\relax (x )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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