Optimal. Leaf size=18 \[ \frac {1}{3} \left (\sin ^2(1-x)+1\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 261
Rule 3198
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 = 0.03, size = 18, normalized size = 1.00 \[ \frac {1}{3} \left (\sin ^2(1-x)+1\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 14, normalized size = 0.78 \[ \frac {1}{3} \, {\left (-\cos \left (x - 1\right )^{2} + 2\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 12, normalized size = 0.67 \[ \frac {1}{3} \, {\left (\sin \left (x - 1\right )^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 13, normalized size = 0.72 \[ \frac {\left (1+\sin ^{2}\left (-1+x \right )\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 12, normalized size = 0.67 \[ \frac {1}{3} \, {\left (\sin \left (x - 1\right )^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.99, size = 12, normalized size = 0.67 \[ \frac {{\left ({\sin \left (x-1\right )}^2+1\right )}^{3/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.75, size = 32, normalized size = 1.78 \[ \frac {\sqrt {\sin ^{2}{\left (x - 1 \right )} + 1} \sin ^{2}{\left (x - 1 \right )}}{3} + \frac {\sqrt {\sin ^{2}{\left (x - 1 \right )} + 1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________