Optimal. Leaf size=14 \[ \frac{1}{3} \left (\sin ^2(x)+1\right )^{3/2} \]
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Rubi [A] time = 0.0341567, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3198, 261} \[ \frac{1}{3} \left (\sin ^2(x)+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3198
Rule 261
Rubi steps
\begin{align*} \int \cos (x) \sin (x) \sqrt{1+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int x \sqrt{1+x^2} \, dx,x,\sin (x)\right )\\ &=\frac{1}{3} \left (1+\sin ^2(x)\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0067136, size = 14, normalized size = 1. \[ \frac{1}{3} \left (\sin ^2(x)+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 11, normalized size = 0.8 \begin{align*}{\frac{1}{3} \left ( 1+ \left ( \sin \left ( 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 = 0.964293, size = 14, normalized size = 1. \begin{align*} \frac{1}{3} \,{\left (\sin \left (x\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 = 2.14527, size = 36, normalized size = 2.57 \begin{align*} \frac{1}{3} \,{\left (-\cos \left (x\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 = 0.877731, size = 27, normalized size = 1.93 \begin{align*} \frac{\sqrt{\sin ^{2}{\left (x \right )} + 1} \sin ^{2}{\left (x \right )}}{3} + \frac{\sqrt{\sin ^{2}{\left (x \right )} + 1}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11555, size = 14, normalized size = 1. \begin{align*} \frac{1}{3} \,{\left (\sin \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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