Optimal. Leaf size=18 \[ \frac{1}{3} \left (\sin ^2(1-x)+1\right )^{3/2} \]
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Rubi [A] time = 0.0411609, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3198, 261} \[ \frac{1}{3} \left (\sin ^2(1-x)+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3198
Rule 261
Rubi steps
\begin{align*} \int -\cos (1-x) \sin (1-x) \sqrt{1+\sin ^2(1-x)} \, dx &=\operatorname{Subst}\left (\int x \sqrt{1+x^2} \, dx,x,\sin (1-x)\right )\\ &=\frac{1}{3} \left (1+\sin ^2(1-x)\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.034236, size = 18, normalized size = 1. \[ \frac{1}{3} \left (\sin ^2(1-x)+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 13, normalized size = 0.7 \begin{align*}{\frac{1}{3} \left ( 1+ \left ( \sin \left ( -1+x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95522, size = 16, normalized size = 0.89 \begin{align*} \frac{1}{3} \,{\left (\sin \left (x - 1\right )^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43615, size = 42, normalized size = 2.33 \begin{align*} \frac{1}{3} \,{\left (-\cos \left (x - 1\right )^{2} + 2\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.856694, size = 32, normalized size = 1.78 \begin{align*} \frac{\sqrt{\sin ^{2}{\left (x - 1 \right )} + 1} \sin ^{2}{\left (x - 1 \right )}}{3} + \frac{\sqrt{\sin ^{2}{\left (x - 1 \right )} + 1}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07729, size = 16, normalized size = 0.89 \begin{align*} \frac{1}{3} \,{\left (\sin \left (x - 1\right )^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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