Optimal. Leaf size=514 \[ -\frac {a^2 \sqrt [4]{-a^2+b^2} g^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} f}-\frac {a^2 \sqrt [4]{-a^2+b^2} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} f}+\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g}+\frac {2 a^3 g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \sqrt {g \cos (e+f x)}}-\frac {2 a g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 b^2 f \sqrt {g \cos (e+f x)}}-\frac {a^3 \left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^3 \left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) f \sqrt {g \cos (e+f x)}}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.79, antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 15, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used =
{2977, 2715, 2721, 2720, 2645, 30, 2774, 2946, 2781, 2886, 2884, 335, 218, 214, 211}
\begin {gather*} \frac {2 a^3 g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \sqrt {g \cos (e+f x)}}-\frac {a^2 g^{3/2} \sqrt [4]{b^2-a^2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{7/2} f}+\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {a^2 g^{3/2} \sqrt [4]{b^2-a^2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{7/2} f}-\frac {a^3 g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {g \cos (e+f x)}}-\frac {a^3 g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {g \cos (e+f x)}}-\frac {2 a g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 b^2 f \sqrt {g \cos (e+f x)}}-\frac {2 a g \sin (e+f x) \sqrt {g \cos (e+f x)}}{3 b^2 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 211
Rule 214
Rule 218
Rule 335
Rule 2645
Rule 2715
Rule 2720
Rule 2721
Rule 2774
Rule 2781
Rule 2884
Rule 2886
Rule 2946
Rule 2977
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx &=\int \left (-\frac {a (g \cos (e+f x))^{3/2}}{b^2}+\frac {(g \cos (e+f x))^{3/2} \sin (e+f x)}{b}+\frac {a^2 (g \cos (e+f x))^{3/2}}{b^2 (a+b \sin (e+f x))}\right ) \, dx\\ &=-\frac {a \int (g \cos (e+f x))^{3/2} \, dx}{b^2}+\frac {a^2 \int \frac {(g \cos (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx}{b^2}+\frac {\int (g \cos (e+f x))^{3/2} \sin (e+f x) \, dx}{b}\\ &=\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f}-\frac {\text {Subst}\left (\int x^{3/2} \, dx,x,g \cos (e+f x)\right )}{b f g}+\frac {\left (a^2 g^2\right ) \int \frac {b+a \sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^3}-\frac {\left (a g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)}} \, dx}{3 b^2}\\ &=\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f}+\frac {\left (a^3 g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)}} \, dx}{b^4}-\frac {\left (a^2 \left (a^2-b^2\right ) g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{b^4}-\frac {\left (a g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{3 b^2 \sqrt {g \cos (e+f x)}}\\ &=\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g}-\frac {2 a g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 b^2 f \sqrt {g \cos (e+f x)}}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f}-\frac {\left (a^3 \sqrt {-a^2+b^2} g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^4}-\frac {\left (a^3 \sqrt {-a^2+b^2} g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^4}-\frac {\left (a^2 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) g^2+b^2 x^2\right )} \, dx,x,g \cos (e+f x)\right )}{b^3 f}+\frac {\left (a^3 g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{b^4 \sqrt {g \cos (e+f x)}}\\ &=\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g}+\frac {2 a^3 g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \sqrt {g \cos (e+f x)}}-\frac {2 a g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 b^2 f \sqrt {g \cos (e+f x)}}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f}-\frac {\left (2 a^2 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^3 f}-\frac {\left (a^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^4 \sqrt {g \cos (e+f x)}}-\frac {\left (a^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^4 \sqrt {g \cos (e+f x)}}\\ &=\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g}+\frac {2 a^3 g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \sqrt {g \cos (e+f x)}}-\frac {2 a g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 b^2 f \sqrt {g \cos (e+f x)}}+\frac {a^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f}-\frac {\left (a^2 \sqrt {-a^2+b^2} g^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^3 f}-\frac {\left (a^2 \sqrt {-a^2+b^2} g^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^3 f}\\ &=-\frac {a^2 \sqrt [4]{-a^2+b^2} g^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} f}-\frac {a^2 \sqrt [4]{-a^2+b^2} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} f}+\frac {2 a^2 g \sqrt {g \cos (e+f x)}}{b^3 f}-\frac {2 (g \cos (e+f x))^{5/2}}{5 b f g}+\frac {2 a^3 g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \sqrt {g \cos (e+f x)}}-\frac {2 a g^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 b^2 f \sqrt {g \cos (e+f x)}}+\frac {a^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {2 a g \sqrt {g \cos (e+f x)} \sin (e+f x)}{3 b^2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 56.20, size = 1953, normalized size = 3.80 \begin {gather*} \frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (-\frac {\cos (2 (e+f x))}{5 b}-\frac {2 a \sin (e+f x)}{3 b^2}\right )}{f}+\frac {(g \cos (e+f x))^{3/2} \left (-\frac {2 \left (10 a^2+3 b^2\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {5 a \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \sqrt {\cos (e+f x)}}{\sqrt {1-\cos ^2(e+f x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )-2 \left (2 b^2 F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)\right ) \left (a^2+b^2 \left (-1+\cos ^2(e+f x)\right )\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )+\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )-\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )\right )}{\left (-a^2+b^2\right )^{3/4}}\right ) \sin (e+f x)}{\sqrt {1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac {\left (30 a^2-3 b^2\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \cos (2 (e+f x)) \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-2 a^2+b^2\right ) \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-2 a^2+b^2\right ) \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}+\frac {4 \sqrt {\cos (e+f x)}}{b}-\frac {4 a F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {5}{2}}(e+f x)}{5 \left (a^2-b^2\right )}+\frac {10 a \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \sqrt {\cos (e+f x)}}{\sqrt {1-\cos ^2(e+f x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )-2 \left (2 b^2 F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)\right ) \left (a^2+b^2 \left (-1+\cos ^2(e+f x)\right )\right )}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (-2 a^2+b^2\right ) \log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (-2 a^2+b^2\right ) \log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}\right ) \sin (e+f x)}{\sqrt {1-\cos ^2(e+f x)} \left (-1+2 \cos ^2(e+f x)\right ) (a+b \sin (e+f x))}+\frac {28 a b \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {5 b \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \sqrt {\cos (e+f x)} \sqrt {1-\cos ^2(e+f x)}}{\left (-5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+2 \left (2 b^2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (a^2-b^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)\right ) \left (a^2+b^2 \left (-1+\cos ^2(e+f x)\right )\right )}+\frac {a \left (-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}\right ) \sin ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) (a+b \sin (e+f x))}\right )}{60 b^2 f \cos ^{\frac {3}{2}}(e+f x)} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 54.24, size = 1067, normalized size = 2.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1067\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^2\,{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{a+b\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________