Optimal. Leaf size=140 \[ -\sqrt {2 \pi } \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\sqrt {1-x} \cos (x) \]
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Rubi [A] time = 0.19, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6129, 6742, 3353, 3352, 3351, 3385, 3354} \[ -\sqrt {2 \pi } \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\sqrt {1-x} \cos (x) \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3354
Rule 3385
Rule 6129
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)} x \sin (x)}{\sqrt {1+x}} \, dx &=\int \frac {x \sin (x)}{\sqrt {1-x}} \, dx\\ &=2 \operatorname {Subst}\left (\int \left (-1+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\sin \left (1-x^2\right )+x^2 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt {1-x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \sin \left (1-x^2\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 )\\ &=\sqrt {1-x} \cos (x)+(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 )-\operatorname {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=\sqrt {1-x} \cos (x)+\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)-\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 {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)-\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 = 8.17, size = 162, normalized size = 1.16 \[ \frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt {x+1} \left ((\cos (x+1)-i \sin (x+1)) \left ((1+2 i) \sqrt {2 \pi } \sqrt {x-1} \text {erf}\left (\frac {(1+i) \sqrt {x-1}}{\sqrt {2}}\right ) (\sin (x)-i \cos (x))-(2-2 i) (x-1) (\cos (1)+i \sin (1))\right )+(-2-i) \sqrt {2 \pi } \sqrt {x-1} \text {erfi}\left (\frac {(1+i) \sqrt {x-1}}{\sqrt {2}}\right ) (\cos (1)+i \sin (1))-(2-2 i) (x-1) (\cos (x)+i \sin (x))\right )}{\sqrt {1-x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} \sqrt {x + 1} x \sin \relax (x)}{x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.52, size = 74, normalized size = 0.53 \[ -\left (\frac {3}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{i} + \left (\frac {3}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{\left (-i\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (-i \, x\right )} + 1.16622538328000 \]
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 \[ \int \frac {\sqrt {1+x}\, x \sin \relax (x )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 347, normalized size = 2.48 \[ -\frac {{\left ({\left ({\left ({\left (-i \, \cos \relax (1) - \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (i \, \cos \relax (1) - \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right ) - {\left ({\left (\cos \relax (1) - i \, \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (\cos \relax (1) + i \, \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} x + {\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 )} {\left | x - 1 \right |} + {\left ({\left (i \, \cos \relax (1) + \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (-i \, \cos \relax (1) + \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right ) + {\left ({\left (\cos \relax (1) - i \, \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (\cos \relax (1) + i \, \sin \relax (1)\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} \sqrt {-x + 1} \sqrt {{\left | x - 1 \right |}}}{2 \, {\left (x - 1\right )}^{2}} \]
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 \[ \int \frac {x\,\sin \relax (x)\,\sqrt {x+1}}{\sqrt {1-x^2}} \,d x \]
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 \[ \int \frac {x \sqrt {x + 1} \sin {\relax (x )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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