Optimal. Leaf size=193 \[ \frac {9}{2} \sqrt {\frac {\pi }{2}} \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\frac {7}{4} \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\frac {7}{4} \sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\frac {9}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\frac {5}{2} (x+1)^{3/2} \sin (x)+\frac {9}{2} \sqrt {x+1} \sin (x)+(x+1)^{5/2} \cos (x)-3 (x+1)^{3/2} \cos (x)-\frac {7}{4} \sqrt {x+1} \cos (x) \]
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Rubi [A] time = 0.40, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 19, 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} \[ \frac {9}{2} \sqrt {\frac {\pi }{2}} \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\frac {7}{4} \sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\frac {7}{4} \sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\frac {9}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\frac {5}{2} (x+1)^{3/2} \sin (x)+\frac {9}{2} \sqrt {x+1} \sin (x)+(x+1)^{5/2} \cos (x)-3 (x+1)^{3/2} \cos (x)-\frac {7}{4} \sqrt {x+1} \cos (x) \]
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)} (1-x)^{3/2} x \sin (x) \, dx &=\int (1-x) x \sqrt {1+x} \sin (x) \, dx\\ &=2 \operatorname {Subst}\left (\int x^2 \left (-2+x^2\right ) \left (-1+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (2 x^2 \sin \left (1-x^2\right )-3 x^4 \sin \left (1-x^2\right )+x^6 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int x^6 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )+4 \operatorname {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-6 \operatorname {Subst}\left (\int x^4 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {1+x} \cos (x)-3 (1+x)^{3/2} \cos (x)+(1+x)^{5/2} \cos (x)-2 \operatorname {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-5 \operatorname {Subst}\left (\int x^4 \cos \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )+9 \operatorname {Subst}\left (\int x^2 \cos \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {1+x} \cos (x)-3 (1+x)^{3/2} \cos (x)+(1+x)^{5/2} \cos (x)+\frac {9}{2} \sqrt {1+x} \sin (x)-\frac {5}{2} (1+x)^{3/2} \sin (x)+\frac {9}{2} \operatorname {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-\frac {15}{2} \operatorname {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-(2 \cos (1)) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )-(2 \sin (1)) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {7}{4} \sqrt {1+x} \cos (x)-3 (1+x)^{3/2} \cos (x)+(1+x)^{5/2} \cos (x)-\sqrt {2 \pi } \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )-\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1)+\frac {9}{2} \sqrt {1+x} \sin (x)-\frac {5}{2} (1+x)^{3/2} \sin (x)+\frac {15}{4} \operatorname {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} (9 \cos (1)) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} (9 \sin (1)) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {7}{4} \sqrt {1+x} \cos (x)-3 (1+x)^{3/2} \cos (x)+(1+x)^{5/2} \cos (x)-\sqrt {2 \pi } \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )-\frac {9}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )+\frac {9}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1)-\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1)+\frac {9}{2} \sqrt {1+x} \sin (x)-\frac {5}{2} (1+x)^{3/2} \sin (x)+\frac {1}{4} (15 \cos (1)) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{4} (15 \sin (1)) \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {7}{4} \sqrt {1+x} \cos (x)-3 (1+x)^{3/2} \cos (x)+(1+x)^{5/2} \cos (x)+\frac {15}{4} \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )-\sqrt {2 \pi } \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )-\frac {9}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )+\frac {9}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1)+\frac {15}{4} \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1)-\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1)+\frac {9}{2} \sqrt {1+x} \sin (x)-\frac {5}{2} (1+x)^{3/2} \sin (x)\\ \end {align*}
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Mathematica [C] time = 8.59, size = 193, normalized size = 1.00 \[ \frac {\left (\frac {1}{32}+\frac {i}{32}\right ) \sqrt {1-x} \left ((\cos (1)-i \sin (1)) \left ((2+2 i) \left (-4 i x^3+10 x^2+(2+19 i) x-(8-15 i)\right ) (\cos (x+1)+i \sin (x+1))-(18-7 i) \sqrt {2 \pi } \sqrt {-x-1} \text {erf}\left (\frac {(1+i) \sqrt {-x-1}}{\sqrt {2}}\right )\right )+(18+7 i) e^i \sqrt {2 \pi } \sqrt {-x-1} \text {erfi}\left (\frac {(1+i) \sqrt {-x-1}}{\sqrt {2}}\right )+(2+2 i) e^{-i x} \left (-4 i x^3-10 x^2-(2-19 i) x+(8+15 i)\right )\right )}{\sqrt {1-x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{2} + 1} x \sqrt {-x + 1} \sin \relax (x), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.22, size = 122, normalized size = 0.63 \[ \left (\frac {25}{32} i + \frac {11}{32}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{i} - \left (\frac {25}{32} i - \frac {11}{32}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{\left (-i\right )} - \frac {1}{8} i \, {\left (4 i \, {\left (x + 1\right )}^{\frac {5}{2}} - \left (12 i + 10\right ) \, {\left (x + 1\right )}^{\frac {3}{2}} - \left (3 i - 18\right ) \, \sqrt {x + 1}\right )} e^{\left (i \, x\right )} - \frac {1}{8} i \, {\left (4 i \, {\left (x + 1\right )}^{\frac {5}{2}} - \left (12 i - 10\right ) \, {\left (x + 1\right )}^{\frac {3}{2}} - \left (3 i + 18\right ) \, \sqrt {x + 1}\right )} e^{\left (-i \, x\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (i \, x\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (-i \, x\right )} - 0.330988710799000 \]
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 \[ \int \frac {\left (1+x \right ) \left (1-x \right )^{\frac {3}{2}} x \sin \relax (x )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.57, size = 1488, normalized size = 7.71 \[ \text {result too large to display} \]
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 \[ \int \frac {x\,\sin \relax (x)\,{\left (1-x\right )}^{3/2}\,\left (x+1\right )}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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