Optimal. Leaf size=240 \[ -2 \sqrt {2 \pi } \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-3 \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-3 \sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {1-x} \sin (x)+(1-x)^{3/2} (-\cos (x))+3 \sqrt {1-x} \cos (x) \]
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Rubi [A] time = 0.40, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6129, 6742, 3353, 3352, 3351, 3385, 3354, 3386} \[ -2 \sqrt {2 \pi } \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-3 \sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-3 \sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {1-x} \sin (x)+(1-x)^{3/2} (-\cos (x))+3 \sqrt {1-x} \cos (x) \]
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)} x \sqrt {1+x} \sin (x) \, dx &=\int \frac {x (1+x) \sin (x)}{\sqrt {1-x}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \left (-2+x^2\right ) \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 (2 \sin \left (1-x^2\right )-3 x^2 \sin \left (1-x^2\right )+x^4 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt {1-x}\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int x^4 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\right )-4 \operatorname {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+6 \operatorname {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \operatorname {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+3 \operatorname {Subst}\left (\int x^2 \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+(4 \cos (1)) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )-(4 \sin (1)) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\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 )-(3 \cos (1)) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )-(3 \sin (1)) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-3 \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 )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \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 )+2 \sqrt {2 \pi } \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)-2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-3 \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 = 9.27, size = 184, normalized size = 0.77 \[ -\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \sqrt {x+1} \left ((\cos (x+1)-i \sin (x+1)) \left ((2+2 i) \left (2 x^2+(2-3 i) x-(4-3 i)\right ) (\cos (1)+i \sin (1))-(6+5 i) \sqrt {2 \pi } \sqrt {x-1} \text {erf}\left (\frac {(1+i) \sqrt {x-1}}{\sqrt {2}}\right ) (\cos (x)+i \sin (x))\right )+(-6+5 i) \sqrt {2 \pi } \sqrt {x-1} \text {erfi}\left (\frac {(1+i) \sqrt {x-1}}{\sqrt {2}}\right ) (\cos (1)+i \sin (1))+(2+2 i) \left (2 x^2+(2+3 i) x-(4+3 i)\right ) (\cos (x)+i \sin (x))\right )}{\sqrt {1-x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} \sqrt {x + 1} x \sin \relax (x)}{x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.20, size = 124, normalized size = 0.52 \[ -\left (\frac {11}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{i} + \left (\frac {11}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{\left (-i\right )} - \frac {1}{4} i \, {\left (-2 i \, {\left (-x + 1\right )}^{\frac {3}{2}} + \left (4 i - 3\right ) \, \sqrt {-x + 1}\right )} e^{\left (i \, x\right )} - \frac {1}{4} i \, {\left (-2 i \, {\left (-x + 1\right )}^{\frac {3}{2}} + \left (4 i + 3\right ) \, \sqrt {-x + 1}\right )} e^{\left (-i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (-i \, x\right )} + 1.79526793396000 \]
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 \[ \int \frac {\left (1+x \right )^{\frac {3}{2}} x \sin \relax (x )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.44, size = 639, normalized size = 2.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\sin \relax (x)\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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