Optimal. Leaf size=236 \[ -4 \sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\frac{3}{2} \sqrt{\frac{\pi }{2}} \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-2 \sqrt{2 \pi } \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-2 \sqrt{2 \pi } \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+4 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\frac{3}{2} \sqrt{\frac{\pi }{2}} \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\frac{3}{2} \sqrt{1-x} \sin (x)+(1-x)^{3/2} (-\cos (x))+4 \sqrt{1-x} \cos (x) \]
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Rubi [A] time = 0.262805, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6129, 6742, 3353, 3352, 3351, 3385, 3354, 3386} \[ -4 \sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\frac{3}{2} \sqrt{\frac{\pi }{2}} \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-2 \sqrt{2 \pi } \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-2 \sqrt{2 \pi } \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+4 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\frac{3}{2} \sqrt{\frac{\pi }{2}} \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\frac{3}{2} \sqrt{1-x} \sin (x)+(1-x)^{3/2} (-\cos (x))+4 \sqrt{1-x} \cos (x) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 6742
Rule 3353
Rule 3352
Rule 3351
Rule 3385
Rule 3354
Rule 3386
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} (1+x)^{3/2} \sin (x) \, dx &=\int \frac{(1+x)^2 \sin (x)}{\sqrt{1-x}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \left (-2+x^2\right )^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (4 \sin \left (1-x^2\right )-4 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 )-8 \operatorname{Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )+8 \operatorname{Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=4 \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 )-4 \operatorname{Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )+(8 \cos (1)) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-(8 \sin (1)) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=4 \sqrt{1-x} \cos (x)-(1-x)^{3/2} \cos (x)+4 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-4 \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 )-(4 \cos (1)) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-(4 \sin (1)) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=4 \sqrt{1-x} \cos (x)-(1-x)^{3/2} \cos (x)-2 \sqrt{2 \pi } \cos (1) C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+4 \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-4 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-2 \sqrt{2 \pi } 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 )\\ &=4 \sqrt{1-x} \cos (x)-(1-x)^{3/2} \cos (x)-2 \sqrt{2 \pi } \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 )+4 \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)-4 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-\frac{3}{2} \sqrt{1-x} \sin (x)\\ \end{align*}
Mathematica [C] time = 8.35558, size = 178, normalized size = 0.75 \[ \frac{i \sqrt{1-x^2} \left (-(\cos (x+1)-i \sin (x+1)) \left ((21+5 i) \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{(1+i) \sqrt{x-1}}{\sqrt{2}}\right ) (\sin (x)-i \cos (x))+2 \sqrt{x-1} (2 i x+(3+6 i)) (\cos (1)+i \sin (1))\right )+(5+21 i) \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\frac{(1+i) \sqrt{x-1}}{\sqrt{2}}\right ) (\sin (1)-i \cos (1))+2 \sqrt{x-1} (2 x+(6+3 i)) (\sin (x)-i \cos (x))\right )}{8 \sqrt{x-1} \sqrt{x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.27, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( x \right ) \left ( 1+x \right ) ^{{\frac{5}{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 = 1.3232, size = 863, normalized size = 3.66 \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 (-\frac{\sqrt{-x^{2} + 1}{\left (x + 1\right )}^{\frac{3}{2}} \sin \left (x\right )}{x - 1}, 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 = 1.26874, size = 165, normalized size = 0.7 \begin{align*} -\left (\frac{21}{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{21}{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 )} + \sqrt{-x + 1} e^{\left (i \, x\right )} + \sqrt{-x + 1} e^{\left (-i \, x\right )} + 3.78811297138 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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