Optimal. Leaf size=405 \[ -\frac {b d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {c^2 x^2+1}}+\frac {d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}-\frac {7 b c d x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^6 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {c^2 x^2+1}}-\frac {7 b^2 d x \sqrt {c^2 d x^2+d}}{1152 c^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {c^2 d x^2+d}+\frac {43 b^2 d x^3 \sqrt {c^2 d x^2+d}}{1728}+\frac {7 b^2 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c^3 \sqrt {c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.66, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5744, 5742, 5758, 5675, 5661, 321, 215, 14, 5730, 12, 459} \[ -\frac {b c^3 d x^6 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {c^2 x^2+1}}-\frac {7 b c d x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {c^2 x^2+1}}+\frac {d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}-\frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {c^2 x^2+1}}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {c^2 d x^2+d}+\frac {43 b^2 d x^3 \sqrt {c^2 d x^2+d}}{1728}-\frac {7 b^2 d x \sqrt {c^2 d x^2+d}}{1152 c^2}+\frac {7 b^2 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c^3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 215
Rule 321
Rule 459
Rule 5661
Rule 5675
Rule 5730
Rule 5742
Rule 5744
Rule 5758
Rubi steps
\begin {align*} \int x^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \int x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{12 \sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx}{36 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{64} b^2 d x^3 \sqrt {d+c^2 d x^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{27 \sqrt {1+c^2 x^2}}\\ &=\frac {b^2 d x \sqrt {d+c^2 d x^2}}{128 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{36 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{128 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{72 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}+\frac {7 b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.19, size = 508, normalized size = 1.25 \[ \frac {-864 a^2 d^{3/2} \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+864 a^2 c d x \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+2304 a^2 c^5 d x^5 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+4032 a^2 c^3 d x^3 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+72 b d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (-12 a-3 b \sinh \left (2 \sinh ^{-1}(c x)\right )+3 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+216 a b d \sqrt {c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-108 a b d \sqrt {c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b d \sqrt {c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )+12 b d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \left (-36 a \sinh \left (2 \sinh ^{-1}(c x)\right )+36 a \sinh \left (4 \sinh ^{-1}(c x)\right )+12 a \sinh \left (6 \sinh ^{-1}(c x)\right )+18 b \cosh \left (2 \sinh ^{-1}(c x)\right )-9 b \cosh \left (4 \sinh ^{-1}(c x)\right )-2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )-288 b^2 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^3-108 b^2 d \sqrt {c^2 d x^2+d} \sinh \left (2 \sinh ^{-1}(c x)\right )+27 b^2 d \sqrt {c^2 d x^2+d} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 d \sqrt {c^2 d x^2+d} \sinh \left (6 \sinh ^{-1}(c x)\right )}{13824 c^3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{2} d x^{4} + a^{2} d x^{2} + {\left (b^{2} c^{2} d x^{4} + b^{2} d x^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{4} + a b d x^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.49, size = 934, normalized size = 2.31 \[ \frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} \arcsinh \left (c x \right ) x^{7}}{3 c^{2} x^{2}+3}+\frac {7 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d}{1152 c^{3} \sqrt {c^{2} x^{2}+1}}+\frac {59 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} x^{5}}{1728 \left (c^{2} x^{2}+1\right )}-\frac {7 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d x}{1152 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {7 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right )}{1152 c^{3} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3} d}{48 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {17 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right )^{2} x^{3}}{48 \left (c^{2} x^{2}+1\right )}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right ) x}{8 c^{2} \left (c^{2} x^{2}+1\right )}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,x^{2}}{16 c \sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} x^{6}}{18 \sqrt {c^{2} x^{2}+1}}-\frac {7 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \,x^{4}}{48 \sqrt {c^{2} x^{2}+1}}+\frac {11 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} \arcsinh \left (c x \right )^{2} x^{5}}{24 \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right )^{2} x}{16 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} \arcsinh \left (c x \right )^{2} x^{7}}{6 c^{2} x^{2}+6}+\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6 c^{2} d}+\frac {11 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} \arcsinh \left (c x \right ) x^{5}}{12 \left (c^{2} x^{2}+1\right )}-\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{16 c^{2} \sqrt {c^{2} d}}+\frac {65 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,x^{3}}{3456 \left (c^{2} x^{2}+1\right )}-\frac {a^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{16 c^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} d}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {17 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right ) x^{3}}{24 \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} \arcsinh \left (c x \right ) x^{6}}{18 \sqrt {c^{2} x^{2}+1}}-\frac {7 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \arcsinh \left (c x \right ) x^{4}}{48 \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right ) x^{2}}{16 c \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} x^{7}}{108 c^{2} x^{2}+108} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________