Optimal. Leaf size=193 \[ \frac{9}{2} \sqrt{\frac{\pi }{2}} \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )+\frac{7}{4} \sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )+\frac{7}{4} \sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )-\frac{9}{2} \sqrt{\frac{\pi }{2}} \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )-\frac{5}{2} (x+1)^{3/2} \sin (x)+\frac{9}{2} \sqrt{x+1} \sin (x)+(x+1)^{5/2} \cos (x)-3 (x+1)^{3/2} \cos (x)-\frac{7}{4} \sqrt{x+1} \cos (x) \]
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Rubi [A] time = 0.40128, antiderivative size = 193, normalized size of antiderivative = 1., 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 6129
Rule 6742
Rule 3385
Rule 3354
Rule 3352
Rule 3351
Rule 3386
Rule 3353
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*}
Mathematica [C] time = 8.27329, size = 193, normalized size = 1. \[ \frac{\left (\frac{1}{32}+\frac{i}{32}\right ) \sqrt{1-x} \left ((\cos (1)-i \sin (1)) \left ((2+2 i) \left (-4 i x^3+10 x^2+(2+19 i) x-(8-15 i)\right ) (\cos (x+1)+i \sin (x+1))-(18-7 i) \sqrt{2 \pi } \sqrt{-x-1} \text{Erf}\left (\frac{(1+i) \sqrt{-x-1}}{\sqrt{2}}\right )\right )+(18+7 i) e^i \sqrt{2 \pi } \sqrt{-x-1} \text{Erfi}\left (\frac{(1+i) \sqrt{-x-1}}{\sqrt{2}}\right )+(2+2 i) e^{-i x} \left (-4 i x^3-10 x^2-(2-19 i) x+(8+15 i)\right )\right )}{\sqrt{1-x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.295, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+x \right ) x\sin \left ( x \right ) \left ( 1-x \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.69048, size = 2009, normalized size = 10.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-x^{2} + 1} x \sqrt{-x + 1} \sin \left (x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.27134, size = 165, normalized size = 0.85 \begin{align*} \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 )} - 0.330988710799 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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